数列・行列関連

定義済み

大SZNO数列システム(LSZNOSs)

\(BW+SZNO(=SpS+SZNO×2)\)

表記

\((a_0,a_1,a_2...a_k)[n]\)

サブルール

• 大小関係

\(A<B⇔∃C[((A<C)∨(A=C))∧(∃c[(\text{ex}(B,c)=C)])]\)

※\(A、B、C\)は列

• 展開

列\(S=(s_0,s_1,s_2...s_m)\)と自然数\(b\)に対して\(\text{ex}(S,b)\)を以下の通り定義する。

• \(q(0)=\begin{cases}Max\{c|s_{c-1}=s_m-2\}&\text{if}　∃c\\0&\text{otherwise}\end{cases}\)
• \(q(c+1)=min\{d|(q(c)<d)∧(s_d=s_m-1)\}\)
• \(l=Max\{c|q(c)=Max\{d|s_d=s_m-1\}\}\)
• \(S_c=(s_{q(c)},s_{q(c)+1},s_{q(c)+2}...s_{q(c+1)-1})\)
• \(S_l=(s_{q(l)},s_{q(l)+1},s_{q(l)+2}...s_m)\)
• \(i=\begin{cases}Max\{q(c)|S_c<S_l\}&\text{if}　∃q(c)\\q(0)&\text{otherwise}\end{cases}\)
• \(A=(s_0,s_1,s_2...s_{i-1})\)
• \(B=(s_i,s_{i+1},s_{i+2}...s_{m-1})\)
• \(\text{ex}(S,b)=A\frown\underbrace{B\frown B\frown B...B}_b\)

計算法

rule1:\((0)[n]=n+1\)

rule2:\((a_0,a_1,a_2...a_{k-1},0)[n]=(a_0,a_1,a_2...a_{k-1})[n+1]\)

rule3:\(\text{ex}((a_0,a_1,a_2...a_k),n)[n]\)

評価

(0,1)=\({\omega}={\beta_0(1)}\)

(0,1,0,0,1)=\({\omega×2}={\beta_0(1,0,1)}\)

(0,1,0,1)=\({\omega^2}={\beta_0(1,1)}\)

(0,1,1)=\({\omega^{\omega}}={\beta_0(2)}\)

(0,1,1,0,1,0,1,1)=\({\omega^{\omega×2}}={\beta_0(2,1,2)}\)

(0,1,1,0,1,1)=\({\omega^{\omega^2}}={\beta_0(2,2)}\)

(0,1,2)=\({\varepsilon_0}={\beta_0({\omega})}={\beta_0({\beta_0(1)})}\)

(0,1,2,1)=\({\psi_0({\omega})}={\beta_0({\beta_0(1,0)})}\)

(0,1,2,1,2)=\({\psi_0({\Omega})}={\beta_0({\beta_0(1,1)})}\)

(0,1,2,2)=\({\psi_0({\Omega^{\omega}})}={\beta_0({\beta_0(2)})}\)

(0,1,2,3)=\({\psi_0({\Omega^{\psi_0(0)}})}={\beta_0({\beta_0({\beta_0(1)})})}\)

(0,1,2,3,...n)=\({\psi_0({\Omega^{\Omega}})}={\beta_0({\Omega})}\)

小偽原始数列システム(SPpSs)

表記

\((a_0,a_1,a_2...a_k)[n]\)

定義

rule1:\((0)[n]=n+1\)

rule2:\((a_0,a_1,a_2...a_{k-1},0)[n]=(a_0,a_1,a_2...a_{k-1})[n+1]\)

rule3:

• \(i=Max\{b|a_b=0\}\)
• \(A=(a_0,a_1,a_2...a_{i-1})\)
• \(B=(a_i,a_{i+1},a_{i+2}...a_{k-1})\)
• \((a_0,a_1,a_2...a_k)[n]=A\frown \underbrace{B\frown B\frown...B}_n[n]\)

評価

(0,1,2,3...)=\({\omega^{\omega}}\)

小偽行列システム(SPMs)

表記

• \(S_x=(a_{x,0},a_{x,1}...a_{x,m})\)
• 正規形:\(S_0S_1S_2...S_k[n]\)

定義

rule1:\(Z[n]=n+1\)

rule2:\(S_0S_1S_2...S_{k-1}Z[n]=S_0S_1S_2...S_{k-1}[n+1]\)

rule3:

• \(i=Max\{b|(b<k)∧(∀c[(0<a_{k,c})⇒(a_{b,c}<a_{k,c})])∧(∃d[(0<a_{k,d})∧(a_{b-1,d}≥a_{b,d})])\}\)
• \({\Delta}=(d_0,d_1,d_2...d_m)\)

• \(d_b=\begin{cases}a_{i,b}-a_{k,b}&\text{if}(0<a_{k,b+1})∧(b<m)\\0&\text{otherwise}\end{cases}\)

• \(A=S_0S_1S_2...S_{i-1}\)
• \(B_0=S_iS_{i+1}S_{i+2}...S_{k-1}\)
• \(B_{b+1}=B_b+{\Delta}\)

\(S_0S_1S_2...S_k[n]=AB_0B_1B_2...B_n[n]\)

評価

(0,0)(1,1)=\({\omega^{\omega}}\)

(0,0)(1,1)(2,0)=|(0,0)(1,1)|(0,0)(1,1)|(0,0)(1,1)|...=\({\omega^{\omega+1}}\)

(0,0)(1,1)(2,0)(3,1)=(0,0)(1,1)|(2,0)|(3,0)|(4,0)|...=\({\omega^{\omega×2}}\)

(0,0)(1,1)(2,2)=|(0,0)(1,1)|(2,0)(3,1)|(4,0)(5,1)|...=\({\omega^{\omega^2}}\)

(0,0)(1,1)(2,2)(3,0)=\({\omega^{\omega^2+1}}\)

(0,0)(1,1)(2,2)(3,0)(4,1)=\({\omega^{\omega^2+{\omega}}}\)

(0,0)(1,1)(2,2)(3,0)(4,1)(5,2)=\({\omega^{\omega^2×2}}\)

(0,0)(1,1)(2,2)(3,3)=\({\omega^{\omega^3}}\)

(0,0,0)(1,1,1)=\({\omega^{\omega^{\omega}}}\)

(0,0,0)(1,1,1)(2,0,0)=\({\omega^{\omega^{\omega}+1}}\)

(0,0,0)(1,1,1)(2,0,0)(3,1,1)=\({\omega^{\omega^{\omega}×2}}\)

(0,0,0)(1,1,1)(2,2,0)=\({\omega^{\omega^{\omega+1}}}\)

(0,0,0)(1,1,1)(2,2,0)(3,3,1)=\({\omega^{\omega^{\omega×2}}}\)

(0,0,0)(1,1,1)(2,2,2)=\({\omega^{\omega^{\omega^2}}}\)

(0,0,0,0)(1,1,1,1)=\({\omega^{\omega^{\omega^{\omega}}}}\)

(0,0,0...0)(1,1,1...1)=\({\varepsilon_0}\)

​大偽原始数列システム(LPpSs)

​表記

\((a_0,a_1,・・・a_k)[n]\)

​計算法

(1)\((\#,0)[n]=(\#)[n+1]\)

(2)\(a_{i-1}≥a_i\)かつ\(i<k\)かつ\(a_i<a_k\)を満たす最大の\(i\)を\(j\)とする（iが存在しない場合、j=0とする）。

• \(A=(a_0,a_1,・・・a_{j-1})\)
• \(B_0=(a_j,a_{j+1},・・・a_{k-1})\)
• \(x=a_k-a_j-1\)
• \(B_m=B_0+(x×m)\)　(\(B_0\)のすべての項に足す)

\((a_0,a_1,・・・a_k)[n]=\{A\frown B_0\frown B_1\frown...B_n\}[n+1]\)

​評価

(0,1)=(0[1])=\({\omega}\)

(0,1,0,1)=(0[1],0[1])=\({\omega×2}\)

(0,1,1)=(0[1],1)=\({\omega^2}\)

(0,1,1,2)=(1[1],1[1])=\({\omega^{\omega}}\)

(0,1,1,2,1)=(0[1],1[1],1)=\({\omega^{\omega+1}}\)

(0,1,1,2,1,2)=(0[1],1[1],1[1])=\({\omega^{\omega×2}}\)

(0,1,1,2,2)=(0[1],1[1],2)=\({\omega^{\omega^2}}\)

(0,1,1,2,2,3)=(0[1],1[1],2[1])=\({\omega^{\omega^{\omega}}}\)

(0,1,2)=(0[1,1])=\({\varepsilon_0}\)

(0,1,2,1)=(0[1,1],1)=\({\varepsilon_0×{\omega}}\)

(0,1,2,1,2,3)=(0[1,1],1[1,1])=\({\varepsilon_0^2}\)

(0,1,2,1,2,3,2)=(0[1,1],1[1,1],2)=\({\varepsilon_0^{\omega}}\)

(0,1,2,1,2,3,2,3,4)=(0[1,1],1[1,1],2[1,1])=\({\varepsilon_0^{\varepsilon_0}}\)

(0,1,2,2)=(0[1,1],2)=\({\varepsilon_1}\)

(0,1,2,2,3)=(0[1,1],2[1])=\({\psi({\omega})}\)

(0,1,2,2,3,4)=(0[1,1],2[1,1])=\({\psi({\psi(0)})}\)

(0,1,2,2,3,4,4,5,6)=(0[1,1],2[1,1],4[1,1])=\({\psi({\psi({\psi(0)})})}\)

(0,1,2,3)=(0[1,1,1])=\({\psi({\Omega})}\)

(0,1,2,・・・n)=(0[1,1,...1])=\({\psi({\Omega^{\omega}})}\)

※限界は(0,2,3,4,・・・n)=(0[2,1,1,...1])=\({\psi({\Omega^{\omega}×2})}\)だと思われる。

新LPpS

表記

\((a_0,a_1,a_2...a_k)[n]\)

定義

rule1:\((0)[n]=n+1\)

rule2:\((a_0,a_1,a_2...a_{k-1},0)[n]=(a_0,a_1,a_2...a_{k-1},0)[n+1]\)

rule3:

• \(i=\begin{cases}k-1&\text{if}　a_k-a_{k-1}=1\\Max\{b|(a_b<a_k)∧(a_{b-1}≥a_b))\}&\text{otherwise}\end{cases}\)
• \(A=(a_0,a_1,a_2...a_{i-1})\)
• \(B_b=(a_i,a_{i+1},a_{i+2}...a_{k-1})+(a_k-a_i)×b\)
• \((a_0,a_1,a_2...a_k)[m]=A\frown B_0\frown B_1\frown B_2...B_n[n]\)

評価

(0,1,2)=(0|1|1|1|...)=\({\omega^{\omega}}\)

(0,1,2,2)=(|0,1,2|1,2,3|2,3,4|...)=\({\varepsilon_0}\)

(0,1,2,2,2)=\({\varepsilon_1}\)

(0,1,2,3)=\({\psi_0({\omega})}\)

(0,1,2,3,2)=\({\psi_0({\omega+1})}\)

(0,1,2,3,2,3,4,5)=\({\psi_0({\psi_0({\omega})})}\)

(0,1,2,3,3)=\({\psi_0({\Omega})}\)

(0,1,2,3,3,2)=\({\psi_0({\Omega+1})}\)

(0,1,2,3,3,2,3,4,5,5)=\({\psi_0({\Omega+{\psi_0({\Omega})}})}\)

(0,1,2,3,3,3)=\({\psi_0({\Omega×2})}\)

(0,1,2,3,4)=\({\psi_0({\Omega×{\omega}})}\)

(0,1,2,3,4,3,4,5,6,7)=\({\psi_0({\Omega×{\psi_0({\Omega×{\omega}})}})}\)

(0,1,2,3,4,4)=\({\psi_0({\Omega^2})}\)

(0,1,2...)=\({\psi_0({\Omega^{\omega}})}\)

大偽行列システム(LPMs)

表記

• \(k,m,n:\)自然数
• \(a_{x,y}:\)非負整数
• \(S_x=(a_{x,0},a_{x,1},...a_{x,m})\)

• 正規形\(:S_0,S_1,...S_k[n]\)

定義

• \(Z:\)零ベクトル
• \(b,c,e,x,y:\)非負整数

rule1:\(Z[n]=n+1\)

rule2:\(S_0S_1...S_{k-1}Z[n]=S_0S_1...S_{k-1}[n+1]\)

rule3:

• \(dim(x)=Max\{b|a_{x,b}>0\}\)
• \(p_0(x)=Max\{b|(b<x)∧(a_{b,0}+1=a_{x,0})\}\)
• \(p_{y+1}(x)=Max\{b|(b<x)∧(a_{b,0}+1=a_{x,0})∧(∃c[p_y^c(x)=b])\}\)
• \(P_0(x)=Max\{b|(b<x)∧(a_{b,0}<a_{x,0})∧((P_0(b)≠b)∨(a_{b,0}=0))\}\)
• \(P_{y+1}(x)=Max\{b|(a_{b,y+1}<a_{x,y+1})∧\)

\(((P_{y+1}(b)≠b-1)∨(a_{b,y+1}=0))∧(∃c[P_y^c(x)=b])\}\)

• \(r=\begin{cases}Max\{b|∀y,y≥dim(k)[∃c[p_y^c(k)=b]]\}&\text{if}　a_{k-1,0}+1=a_{k,0}\\\\Max\{b|(∀y,y≥dim(k)[∃c[P_y^c(k)=b]])∨\\((∀y,y≥dim(k)[∃c[p_y^c(k)=b]])∧(∃c[(c>dim(k))∧\\(p_c(b)≠b-1)∧(a_{b,c}>0)]))\}&\text{otherwise}\end{cases}\)
• \({\Delta}=D_0D_1...D_{k-1-r}\)

• \(D_x=(d_{x,0},d_{x,1},...d_{x,m})\)

• \(d_{0,y}=\begin{cases}a_{k,y}-a_{r,y}&\text{if}　y<dim(k)\\a_{k,y}-a_{r,y}-1&\text{if}　y=dim(k)\\0&\text{otherwise}\end{cases}\)
• \(d_{x+1,y}=\begin{cases}d_{0,y}&\text{if}　∃b[P_y^b(r+x+1)=r]\\0&\text{otherwise}\end{cases}\)

• \(A=S_0S_1...S_{r-1}\)
• \(B_0=S_rS_{r+1}...S_{k-1}\)
• \(B_b=B_0+b×{\Delta}\)

\(S_0S_1...S_k[n]=AB_0B_1...B_n[n]\)

評価

(0,0)(1,1)\(={\psi_0({\Omega^{\omega}})}\)

(0,0)(1,1)(1,1)=|(0,0)(1,1)|(1,0)(2,1)|(2,0)(3,1)|...\(={\psi_0({\Omega^{\omega}+1})}\)

(0,0)(1,1)(2,0)=(0,0)|(1,1)|(1,1)|(1,1)|...\(={\psi_0({\Omega^{\omega}+{\omega}})}\)

(0,0)(1,1)(2,0)(2,0)=|(0,0)(1,1)(2,0)|(1,0)(2,1)(3,0)|(2,0)(3,1)(4,0)|...\(={\psi_0({\Omega^{\omega}+{\omega+1}})}\)

(0,0)(1,1)(2,0)(3,1)=(0,0)(1,1)|(2,0)|(3,0)|(4,0)|...\(={\psi_0({\Omega^{\omega}×2})}\)

(0,0)(1,1)(2,1)\(={\psi_0({\Omega^{\omega}×{\omega}})}\)

(0,0)(1,1)(2,1)(2,1)\(={\psi_0({\Omega^{\omega}×{\omega}+{\Omega}})}\)

(0,0)(1,1)(2,1)(3,0)\(={\psi_0({\Omega^{\omega}×{\omega}+{\Omega×{\omega}}})}\)

(0,0)(1,1)(2,1)(3,0)(4,0)\(={\psi_0({\Omega^{\omega}×{\omega}+{\Omega×{\omega^{\omega}}}})}\)

(0,0)(1,1)(2,1)(3,0)(4,0)(3,1)\(={\psi_0({\Omega^{\omega}×{\omega}+{\Omega^2}})}\)

(0,0)(1,1)(2,1)(3,0)(4,0)(3,1)(2,1)(3,0)(4,0)(3,1)\(={\psi_0({\Omega^{\omega}×{\omega}+{\Omega^2×2}})}\)

(0,0)(1,1)(2,1)(3,0)(4,0)(3,1)(3,1)\(={\psi_0({\Omega^{\omega}×{\omega}+{\Omega^3}})}\)

(0,0)(1,1)(2,1)(3,0)(4,0)(3,1)(4,0)\(={\psi_0({\Omega^{\omega}×({\omega+1})})}\)

(0,0)(1,1)(2,1)(3,0)(4,0)(3,1)(4,0)(3,1)\(={\psi_0({\Omega^{\omega+1}})}\)

(0,0)(1,1)(2,1)(3,0)(4,0)(3,1)(4,0)(5,0)(4,1)\(={\psi_0({\Omega^{\Omega}})}\)

(0,0)(1,1)(2,1)(3,0)(4,0)(4,0)\(={\psi_0({\psi_1(0)})}\)

(0,0)(1,1)(2,1)(3,0)(4,0)(5,0)\(={\psi_0({\psi_1({\omega})})}\)

(0,0)(1,1)(2,1)(3,0)(4,0)(5,0)(4,1)\(={\psi_0({\psi_1({\Omega})})}\)

(0,0)(1,1)(2,1)(3,0)(4,0)(5,0)(4,1)(5,0)(6,0)(6,0)\(={\psi_0({\psi_1({\psi_1(0)})})}\)

(0,0)(1,1)(2,1)(3,0)(4,0)(5,0)(5,0)\(={\psi_0({\psi_1({\Omega_2})})}\)

(0,0)(1,1)(2,1)(3,0)(4,1)\(={\psi_0({\psi_1({\Omega_2^{\omega}})})}\)

(0,0)(1,1)(2,1)(3,0)(4,1)(4,0)\(={\psi_0({\psi_1({\Omega_2^{\omega}+1})})}\)

(0,0)(1,1)(2,1)(3,0)(4,1)(4,1)\(={\psi_0({\psi_1({\Omega_2^{\omega}+{\Omega}})})}\)

(0,0)(1,1)(2,1)(3,0)(4,1)(5,0)(4,1)(5,0)(6,1)\(={\psi_0({\psi_1({\Omega_2^{\omega}+{\psi_1({\Omega_2^{\omega}})}})})}\)

(0,0)(1,1)(2,1)(3,0)(4,1)(5,0)(5,0)\(={\psi_0({\psi_1({\Omega_2^{\omega}+{\Omega_2}})})}\)

(0,0)(1,1)(2,1)(3,0)(4,1)(5,1)\(={\psi_0({\psi_1({\Omega_2^{\omega}×{\omega}})})}\)

(0,0)(1,1)(2,1)(3,0)(4,1)(5,1)(4,0)\(={\psi_0({\psi_1({\Omega_2^{\omega}×{\omega}+1})})}\)

(0,0)(1,1)(2,1)(3,0)(4,1)(5,1)(4,1)\(={\psi_0({\psi_1({\Omega_2^{\omega}×{\omega}+{\Omega}})})}\)

(0,0)(1,1)(2,1)(3,0)(4,1)(5,1)(5,0)\(={\psi_0({\psi_1({\Omega_2^{\omega}×{\omega}+{\Omega_2}})})}\)

(0,0)(1,1)(2,1)(3,0)(4,1)(5,1)(5,1)\(={\psi_0({\psi_1({\Omega_2^{\omega}×{\omega}+{\Omega_2×{\Omega}}})})}\)

(0,0)(1,1)(2,1)(3,0)(4,1)(5,1)(6,0)\(={\psi_0({\psi_1({\Omega_2^{\omega}×{\omega}+{\Omega_2×{\Omega×{\omega}}}})})}\)

(0,0)(1,1)(2,1)(3,0)(4,1)(5,1)(6,0)(7,0)(7,0)\(={\psi_0({\psi_1({\Omega_2^{\omega}×{\omega}+{\Omega_2×{\psi_1(0)}}})})}\)

(0,0)(1,1)(2,1)(3,0)(4,1)(5,1)(6,0)(7,1)(8,1)\(={\psi_0({\psi_1({\Omega_2^{\omega}×{\omega}+{\Omega_2×{\psi_1({\Omega_2^{\omega}×{\omega}})}}})})}\)

(0,0)(1,1)(2,1)(3,1)\(={\psi_0({\psi_1({\Omega_2^{\omega}×{\omega}+{\Omega_2^2}})})}\)

(0,0)(1,1)(2,1)(3,1)(2,1)(3,1)=|(0,0)(1,1)(2,1)(3,1)(2,1)|(3,0)(4,1)(5,1)(6,1)(5,1)|(6,0)(7,1)(8,1)(9,1)(8,1)|...

(0,0)(1,1)(2,1)(3,1)(2,1)(3,0)(4,1)(5,1)(6,1)\(={\psi_0({\psi_1({\Omega_2^{\omega}×{\omega}+{\Omega_2^2}})×2})}\)

(0,0)(1,1)(2,1)(3,1)(2,1)(3,0)(4,1)(5,1)(6,1)(4,0)\(={\psi_0({\psi_1({\Omega_2^{\omega}×{\omega}+{\Omega_2^2+1}})})}\)

(0,0)(1,1)(2,1)(3,1)(2,1)(3,0)(4,1)(5,1)(6,1)(4,1)(5,0)(6,1)(7,1)(8,1)

\(={\psi_0({\psi_1({\Omega_2^{\omega}×{\omega}+{\Omega_2^2}+{\psi_1({\Omega_2^{\omega}×{\omega}+{\Omega_2^2}})}})})}\)

(0,0)(1,1)(2,1)(3,1)(2,1)(3,0)(4,1)(5,1)(6,1)(5,0)\(={\psi_0({\psi_1({\Omega_2^{\omega}×{\omega}+{\Omega_2^2+{\Omega_2}}})})}\)

(0,0)(1,1)(2,1)(3,1)(2,1)(3,1)\(={\psi_0({\psi_1({\Omega_2^{\omega}×{\omega}+{\Omega_2^2×2}})})}\)

(0,0)(1,1)(2,1)(3,1)(3,0)\(={\psi_0({\psi_1({\Omega_2^{\omega}×{\omega}+{\Omega_2^2×{\omega}}})})}\)

(0,0)(1,1)(2,1)(3,1)(4,0)\(={\psi_0({\psi_1({\Omega_2^{\omega}×{\omega}+{\Omega_2^2×{\Omega^{\omega}}}})})}\)

(0,0)(1,1)(2,1)(3,1)(4,0)(5,1)(6,1)(7,1)\(={\psi_0({\psi_1({\Omega_2^{\omega}×{\omega}+{\Omega_2^2×{\psi_1({\Omega_2^{\omega}×{\omega}+{\Omega_2^2}})}}})})}\)

(0,0)(1,1)(2,1)(3,1)(4,1)\(={\psi_0({\psi_1({\Omega_2^{\omega}×{\omega}+{\Omega_2^3}})})}\)

(0,0)(1,1)(2,1)(3,1)(4,1)(2,1)(3,1)(4,1)\(={\psi_0({\psi_1({\Omega_2^{\omega}×{\omega}+{\Omega_2^3×2}})})}\)

(0,0)(1,1)(2,1)(3,1)(4,1)(3,1)(4,1)\(={\psi_0({\psi_1({\Omega_2^{\omega}×{\omega}+{\Omega_2^4}})})}\)

(0,0)(1,1)(2,1)(3,1)(4,1)(4,0)\(={\psi_0({\psi_1({\Omega_2^{\omega}×({\omega+1})})})}\)

(0,0)(1,1)(2,1)(3,1)(4,1)(4,0)(3,1)(4,1)\(={\psi_0({\psi_1({\Omega_2^{\omega+1}})})}\)

(0,0)(1,1)(2,1)(3,1)(4,1)(4,1)\(={\psi_0({\psi_1({\Omega_2^{\Omega}})})}\)

(0,0)(1,1)(2,1)(3,1)(4,1)(5,0)(6,0)(6,0)\(={\psi_0({\psi_1({\Omega_2^{\psi_1(0)}})})}\)

(0,0)(1,1)(2,1)(3,1)(4,1)(5,1)\(={\psi_0({\psi_1({\Omega_2^{\Omega_2}})})}\)

(0,0)(1,1)(2,2)\(={\psi_0({\psi_1({\psi_2(0)})})}\)

(0,0)(1,1)(2,2)(2,2)\(=|(0,0)(1,1)(2,2)|(2,1)(3,2)(4,3)|(4,2)(5,3)(6,4)|...\)

(0,0)(1,1)(2,2)(2,1)(3,1)\(={\psi_0({\psi_1({\psi_2(0)}+{\Omega_2^2})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)\(={\psi_0({\psi_1({\psi_2(0)×2})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(3,1)(4,2)\(={\psi_0({\psi_1({\psi_2(0)^2})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(3,2)\(={\psi_0({\psi_1({\psi_2(1)})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,0)\(={\psi_0({\psi_1({\psi_2({\omega})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,0)(4,0)\(={\psi_0({\psi_1({\psi_2({\omega+1})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,0)(5,0)(4,1)\(={\psi_0({\psi_1({\psi_2({\Omega})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,0)(5,0)(4,1)(5,2)\(={\psi_0({\psi_1({\psi_2({\psi_1({\psi_2(0)})})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,0)(5,0)(5,0)\(={\psi_0({\psi_1({\psi_2({\Omega_2})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,0)(5,1)\(={\psi_0({\psi_1({\psi_2({\Omega_2^{\omega}})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,0)(5,1)(6,1)\(={\psi_0({\psi_1({\psi_2({\Omega_2^{\omega}×{\omega}})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,1)\(={\psi_0({\psi_1({\psi_2({\Omega_2^{\omega}×{\omega}+{\Omega_2^2}})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,1)(5,2)\(={\psi_0({\psi_1({\psi_2({\psi_2(0)})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)\(={\psi_0({\psi_1({\psi_2({\Omega_3})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(2,1)\(={\psi_0({\psi_1({\psi_2({\Omega_3})})}+{\Omega})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(2,1)(3,0)(4,1)(5,2)(5,1)(6,2)(7,2)\(={\psi_0({\psi_1({\psi_2({\Omega_3})})}×2)}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(2,1)(3,0)(4,1)(5,2)(5,1)(6,2)(7,2)(4,0)\(={\psi_0({\psi_1({\psi_2({\Omega_3})}+1)})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(2,1)(3,0)(4,1)(5,2)(5,1)(6,2)(7,2)(4,1)\(={\psi_0({\psi_1({\psi_2({\Omega_3})}+{\Omega})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(2,1)(3,0)(4,1)(5,2)(5,1)(6,2)(7,2)(5,0)\(={\psi_0({\psi_1({\psi_2({\Omega_3})}+{\Omega_2})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(2,1)(3,1)\(={\psi_0({\psi_1({\psi_2({\Omega_3})}+{\Omega_2^2})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(2,1)(3,2)(4,2)\(={\psi_0({\psi_1({\psi_2({\Omega_3})×2})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(3,1)\(={\psi_0({\psi_1({\psi_2({\Omega_3})}×{\Omega})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(3,1)(4,0)(5,1)(6,2)(6,1)(7,2)(8,2)\(={\psi_0({\psi_1({\psi_2({\Omega_3})}×{\psi_1({\psi_2({\Omega_3})})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(3,1)(4,1)\(={\psi_0({\psi_1({\psi_2({\Omega_3})}×{\Omega_2})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(3,1)(4,2)(5,2)\(={\psi_0({\psi_1({\psi_2({\Omega_3})^2})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(4,0)\(={\psi_0({\psi_1({\psi_2({\Omega_3+1})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(4,1)\(={\psi_0({\psi_1({\psi_2({\Omega_3+{\Omega}})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(4,1)(5,2)\(={\psi_0({\psi_1({\psi_2({\Omega_3+{\Omega_2}})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(4,2)\(={\psi_0({\psi_1({\psi_2({\Omega_3×2})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(5,0)\(={\psi_0({\psi_1({\psi_2({\Omega_3×{\omega}})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(5,0)(6,0)(5,1)\(={\psi_0({\psi_1({\psi_2({\Omega_3×{\Omega}})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(5,0)(6,0)(5,1)(6,0)(7,1)(8,2)(8,1)(9,2)(10,2)\(={\psi_0({\psi_1({\psi_2({\Omega_3×{\psi_1({\psi_2({\Omega_3})})}})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(5,0)(6,0)(5,1)(6,1)\(={\psi_0({\psi_1({\psi_2({\Omega_3×{\Omega_2}})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(5,0)(6,0)(5,1)(6,2)(6,1)(7,2)(8,2)\(={\psi_0({\psi_1({\psi_2({\Omega_3×{\psi_2({\Omega_3})}})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(5,0)(6,0)(5,2)\(={\psi_0({\psi_1({\psi_2({\Omega_3^2})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(5,0)(6,0)(6,0)\(={\psi_0({\psi_1({\psi_2({\psi_3(0)})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(5,0)(6,1)\(={\psi_0({\psi_1({\psi_2({\psi_3({\Omega_4^{\omega}})})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(5,0)(6,1)(7,1)\(={\psi_0({\psi_1({\psi_2({\psi_3({\Omega_4^{\omega}×{\omega}})})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(5,0)(6,1)(7,1)(7,0)\(={\psi_0({\psi_1({\psi_2({\psi_3({\Omega_4^{\omega}×{\omega}+{\Omega_4}})})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(5,0)(6,1)(7,1)(7,1)\(={\psi_0({\psi_1({\psi_2({\psi_3({\Omega_4^{\omega}×{\omega}+{\Omega_4×{\Omega}}})})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(5,0)(6,1)(7,2)\(={\psi_0({\psi_1({\psi_2({\psi_3({\Omega_4^{\omega}+{\Omega_4×{\psi_1({\psi_2(0)})}}})})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(5,1)\(={\psi_0({\psi_1({\psi_2({\psi_3({\Omega_4^{\omega}×{\omega}+{\Omega_4×{\Omega_2}}})})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(5,1)(4,2)\(={\psi_0({\psi_1({\psi_2({\psi_3({\Omega_4^{\omega}×{\omega}+{\Omega_4×{\Omega_2}}})}+{\Omega_3})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(5,1)(4,2)(5,1)\(={\psi_0({\psi_1({\psi_2({\psi_3({\Omega_4^{\omega}×{\omega}+{\Omega_4×{\Omega_2}}})×2})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(5,1)(6,0)\(={\psi_0({\psi_1({\psi_2({\psi_3({\Omega_4^{\omega}×{\omega}+{\Omega_4×{\Omega_2}}})}×{\Omega^{\omega}})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(5,1)(6,1)\(={\psi_0({\psi_1({\psi_2({\psi_3({\Omega_4^{\omega}×{\omega}+{\Omega_4×{\Omega_2}}})×{\Omega_2}})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(5,1)(6,2)\(={\psi_0({\psi_1({\psi_2({\psi_3({\Omega_4^{\omega}×{\omega}+{\Omega_4×{\Omega_2}}})}×{\psi_2(0)})})})}\)

(0,0)(1,1)(2,2)(2,1)(3,2)(4,2)(5,1)(6,2)(7,2)(8,0)(9,1)

\(={\psi_0({\psi_1({\psi_2({\psi_3({\Omega_4^{\omega}×{\omega}+{\Omega_4×{\Omega_2}}})}×{\psi_2({\psi_3({\Omega_4^{\omega}})})})})})}\)

(0,0)(1,1)(2,2)(2,2)\(={\psi_0({\Omega_{\omega}})}=B(0,0,0)(1,1,1)\)

(0,0)(1,1)(2,2)(3,0)\(={\psi_0({\Omega_{\omega}×{\omega}})}\)

(0,0)(1,1)(2,2)(3,2)\(={\psi_0({\Omega_{\omega}×{\omega+1}})}\)

(0,0)(1,1)(2,2)(3,2)(4,0)\(={\psi_0({\Omega_{\omega}×{\omega×2}})}\)

(0,0)(1,1)(2,2)(3,2)(4,0)(3,2)\(={\psi_0({\Omega_{\omega}×({\omega×2+1})})}\)

(0,0)(1,1)(2,2)(3,2)(4,0)(5,0)(5,0)=(0,0)(1,1)(2,2)|(3,2)(4,0)(5,0)|(4,2)(5,0)(6,0)|(6,2)(7,0)(8,0)|...

(0,0)(1,1)(2,2)(3,2)(4,0)(5,0)(4,1)\(={\psi_0({\Omega_{\omega}×{\Omega}})}\)

(0,0)(1,1)(2,2)(3,2)(4,0)(5,0)(4,1)(3,2)=|(0,0)(1,1)(2,2)(3,2)(4,0)(5,0)(4,1)|(3,1)(4,2)(5,3)(6,3)(7,0)(8,0)(7,2)|...

\(={\psi_0({\Omega_{\omega}^2})}=\)B(0,0,0)(1,1,1)(2,1,0)(1,1,1)

(0,0)(1,1)(2,2)(3,2)(4,0)(5,0)(4,2)=B(0,0,0)(1,1,1)(2,1,1)

(0,0)(1,1)(2,2)(3,2)(4,0)(5,0)(5,0)=B(0,0,0)(1,1,1)(2,2,0)

(0,0)(1,1)(2,2)(3,2)(4,0)(5,1)=B(0,0,0)(1,1,1)(2,2,0)(3,2,0)(4,0,0)

m-有限行列システム(m-FMs)

表記

• \(k,m,n:\)自然数
• \(a_{x,y}:\)非負整数
• \(S_x=(a_{x,0},a_{x,1},...a_{x,m})\)

• 正規形\(:S_0,S_1,...S_k[n]\)

定義

• \(Z:\)零ベクトル
• \(b,c,e,x,y:\)非負整数

rule1:\(Z[n]=n+1\)

rule2:\(S_0S_1...S_{k-1}Z[n]=S_0S_1...S_{k-1}[n+1]\)

rule3:

• \(dim(x)=Max\{b|a_{x,b}>0\}\)
• \(P_0(x)=Max\{b|(b<x)∧(a_{b,0}<a_{x,0})∧((P_0(b)≠b)∨(a_{b,0}=0))\}\)
• \(P_{y+1}(x)=Max\{b|(a_{b,y+1}<a_{x,y+1})∧\)

\(((P_{y+1}(b)≠b-1)∨(a_{b,y+1}=0))∧(∃c[P_y^c(x)=b])\}\)

• \(r=Max\{b|∀y,y≥dim(k)[∃c[P_y^c(k)=b]]\}\)
• \({\Delta}=D_0D_1...D_{k-1-r}\)

• \(D_x=(d_{x,0},d_{x,1},...d_{x,m})\)

• \(d_{0,y}=\begin{cases}a_{k,y}-a_{r,y}&\text{if}　y<dim(k)\\a_{k,dim(k)}-a_{r,dim(k)}-1&\text{if}　y≥dim(k)\\0&\text{otherwise}\end{cases}\)
• \(d_{x+1,y}=\begin{cases}d_{0,y}&\text{if}　∃b[P_y^b(r+x+1)=r]\\0&\text{otherwise}\end{cases}\)

• \(A=S_0S_1...S_{r-1}\)
• \(B_0=S_rS_{r+1}...S_{k-1}\)
• \(B_b=B_0+b×{\Delta}\)

\(S_0S_1...S_k[n]=AB_0B_1...B_n[n]\)

評価

• m=1

(0,0)(1,0)(2,0)(3,0)=|(0,0)(1,0)(2,0)|(2,2)(3,0)(4,0)|(4,4)(5,0)(6,0)|...

(0,0)(1,0)(2,0)(2,2)(3,0)(4,0)=(0,0)(1,0)(2,0)|(2,2)(3,0)|(3,3)(4,0)|(4,4)(5,0)|...

(0,0)(1,0)(2,0)(2,2)(3,0)(3,2)=|(0,0)(1,0)(2,0)(2,2)(3,0)|(3,1)(4,0)(5,0)(5,3)(6,0)|(6,2)(7,0)(8,0)(8,4)(9,0)|...

(0,0)(1,0)(2,0)(2,2)=|(0,0)(1,0)(2,0)|(2,1)(3,0)(4,0)|(4,2)(5,0)(6,0)|...

(0,0)(1,0)(2,0)(2,1)(3,0)(4,0)=(0,0)(1,0)(2,0)|(2,1)(3,0)|(3,2)(4,0)|(4,3)(5,0)|...

(0,0)(1,0)(2,0)=|(0,0)(1,0)|(1,1)(2,0)|(2,2)(3,0)|...=B(0,0,0)(1,1,1)

(0,0)(1,0)(2,0)(2,0)(3,0)=B(0,0,0)(1,1,1)(2,0,0)

(0,0)(1,0)(2,0)(2,1)=B(0,0,0)(1,1,1)(2,1,0)

(0,0)(1,0)(2,0)(2,1)(2,1)=B(0,0,0)(1,1,1)(2,1,0)(1,1,1)(2,1,0)

(0,0)(1,0)(2,0)(2,1)(3,0)=B(0,0,0)(1,1,1)(2,1,0)(2,0,0)

(0,0)(1,0)(2,0)(2,1)(3,0)(3,1)=B(0,0,0)(1,1,1)(2,1,0)(2,1,0)

(0,0)(1,0)(2,0)(2,1)(3,0)(3,1)(4,0)=B(0,0,0)(1,1,1)(2,1,0)(3,0,0)

(0,0)(1,0)(2,0)(2,1)(3,0)(3,2)=B(0,0,0)(1,1,1)(2,1,0)(3,2,0)

(0,0)(1,0)(2,0)(2,1)(3,0)(4,0)=B(0,0,0)(1,1,1)(2,1,0)(3,2,1)

(0,0)(1,0)(2,0)(2,2)=B(0,0,0)(1,1,1)(2,1,1)

(0,0)(1,0)(2,0)(2,2)(2,0)=B(0,0,0)(1,1,1)(2,1,1)(1,1,1)

(0,0)(1,0)(2,0)(2,2)(2,1)=B(0,0,0)(1,1,1)(2,1,1)(1,1,1)(2,1,0)

(0,0)(1,0)(2,0)(2,2)(2,2)=B(0,0,0)(1,1,1)(2,1,1)(1,1,1)(2,1,1)

(0,0)(1,0)(2,0)(2,2)(3,0)=B(0,0,0)(1,1,1)(2,1,1)(2,0,0)

(0,0)(1,0)(2,0)(2,2)(3,0)(3,1)=B(0,0,0)(1,1,1)(2,1,1)(2,1,0)

(0,0)(1,0)(2,0)(2,2)(3,0)(3,2)=B(0,0,0)(1,1,1)(2,1,1)(2,1,1)

(0,0)(1,0)(2,0)(2,2)(3,0)(3,3)=B(0,0,0)(1,1,1)(2,2,0)

(0,0)(1,0)(2,0)(2,2)(3,0)(4,0)=B(0,0,0)(1,1,1)(2,2,0)(3,3,1)

(0,0)(1,0)(2,0)(3,0)=B(0,0,0)(1,1,1)(2,2,1)

(0,0)(1,0)(2,0)(3,0)(2,2)(3,0)(4,0)(5,0)=B(0,0,0)(1,1,1)(2,2,1)(1,1,1)(2,2,0)(3,3,1)

(0,0)(1,0)(2,0)(3,0)(3,0)=B(0,0,0)(1,1,1)(2,2,1)(1,1,1)(2,2,1)

(0,0)(1,0)(2,0)(3,0)(3,1)=B(0,0,0)(1,1,1)(2,2,1)(2,1,0)

(0,0)(1,0)(2,0)(3,0)(3,1)(3,0)=B(0,0,0)(1,1,1)(2,2,1)(2,1,0)(1,1,1)(2,2,1)

(0,0)(1,0)(2,0)(3,0)(3,1)(4,0)=B(0,0,0)(1,1,1)(2,2,1)(2,1,0)(2,0,0)

(0,0)(1,0)(2,0)(3,0)(3,1)(4,0)(4,1)=B(0,0,0)(1,1,1)(2,2,1)(2,1,0)(2,1,0)

(0,0)(1,0)(2,0)(3,0)(3,1)(4,0)(5,0)(6,0)=B(0,0,0)(1,1,1)(2,2,1)(2,1,0)(3,2,1)(4,3,1)

(0,0)(1,0)(2,0)(3,0)(3,2)=B(0,0,0)(1,1,1)(2,2,1)(2,1,1)

大偽大偽行列システム(LPLPMs)

表記

• \(k,m,n:\)自然数
• \(a_{x,y}:\)非負整数
• \(S_x=(a_{x,0},a_{x,1},...a_{x,m})\)

• 正規形\(:S_0,S_1,...S_k[n]\)

定義

• \(Z:\)零ベクトル
• \(b,c,e,x,y:\)非負整数

rule1:\(Z[n]=n+1\)

rule2:\(S_0S_1...S_{k-1}Z[n]=S_0S_1...S_{k-1}[n+1]\)

rule3:

• \(dim(x)=Max\{b|a_{x,b}>0\}\)
• \(h=Max\{dim(b)\}\)
• \(p_0(x)=Max\{b|(b<x)∧(a_{b,0}+1=a_{x,0})\}\)
• \(p_{y+1}(x)=Max\{b|(b<x)∧(a_{b,0}+1=a_{x,0})∧(∃c[p_y^c(x)=b])\}\)
• \(P_0(x)=Max\{b|(b<x)∧(a_{b,0}<a_{x,0})∧((P_0(b)≠b)∨(a_{b,0}=0))\}\)
• \(P_{y+1}(x)=Max\{b|(a_{b,y+1}<a_{x,y+1})∧\)

\(((P_{y+1}(b)≠b-1)∨(a_{b,y+1}=0))∧(∃c[P_y^c(x)=b])\}\)

• \(r=\begin{cases}Max\{b|∀y,y≥dim(k)[∃c[p_y^c(k)=b]]\}&\text{if}　a_{k-1,0}+1=a_{k,0}\\Max\{b|∀y,y≥dim(k)[∃c[P_y^c(k)=b]])\}&\text{otherwise}\end{cases}\)
• \({\Delta}=D_0D_1...D_{k-1-r}\)

• \(D_x=(d_{x,0},d_{x,1},...d_{x,m})\)

• \(d_{0,y}=\begin{cases}a_{k,y}-a_{r,y}&\text{if}　y<dim(k)\\a_{k,y}-a_{r,y}-1&\text{if}　dim(k)≤y≤h\\0&\text{otherwise}\end{cases}\)
• \(d_{x+1,y}=\begin{cases}d_{0,y}&\text{if}　∃b[P_y^b(r+x+1)=r]\\0&\text{otherwise}\end{cases}\)

• \(A=S_0S_1...S_{r-1}\)
• \(B_0=S_rS_{r+1}...S_{k-1}\)
• \(B_b=B_0+b×{\Delta}\)

\(S_0S_1...S_k[n]=AB_0B_1...B_n[n]\)

巨大不明行列システム(HUMs)

表記

• \(k,m,n:\)自然数
• \(a_{x,y}:\)非負整数
• \(S_x=(a_{x,0},a_{x,1},...)\)

• 正規形\(:S_0,S_1,...S_k[n]\)

定義

• \(Z:\)零ベクトル
• \(b,c,e,x,y:\)非負整数

rule1:\(Z[n]=n+1\)

rule2:\(S_0S_1...S_{k-1}Z[n]=S_0S_1...S_{k-1}[n+1]\)

rule3:

• \(dim(x)=Max\{b|a_{x,b}>0\}\)
• \(p_0(x)=Max\{b|(b<x)∧(a_{b,0}+1=a_{x,0})\}\)
• \(p_{y+1}(x)=Max\{b|(b<x)∧(a_{b,0}+1=a_{x,0})∧(∃c[p_y^c(x)=b])\}\)
• \(P_0(x)=Max\{b|(b<x)∧(a_{b,0}<a_{x,0})∧((P_0(b)≠b)∨(a_{b,0}=0))\}\)
• \(P_{y+1}(x)=Max\{b|(a_{b,y+1}<a_{x,y+1})∧\)

\(((P_{y+1}(b)≠b-1)∨(a_{b,y+1}=0))∧(∃c[P_y^c(x)=b])\}\)

• \(r=\begin{cases}Max\{b|∀y,y≥dim(k)[∃c[p_y^c(k)=b]]\}&\text{if}　a_{k-1,0}+1=a_{k,0}\\Max\{b|∀y,y≥dim(k)[∃c[P_y^c(k)=b]]\}&\text{otherwise}\end{cases}\)
• \({\Delta}=D_0D_1...D_{k-1-r}\)

• \(D_x=(d_{x,0},d_{x,1},...d_{x,m})\)

• \(d_{0,y}=\begin{cases}a_{k,y}-a_{r,y}&\text{if}　y<dim(k)\\a_{k,y}-a_{r,y}-1&\text{if}　y=dim(k)\\a_{k,y-1}-a_{r,y-1}-1&\text{if}　y=dim(k)+1\\0&\text{otherwise}\end{cases}\)
• \(d_{x+1,y}=\begin{cases}d_{0,y}&\text{if}　∃b[P_y^b(r+x+1)=r]\\0&\text{otherwise}\end{cases}\)

• \(A=S_0S_1...S_{r-1}\)
• \(B_0=S_rS_{r+1}...S_{k-1}\)
• \(B_b=B_0+b×{\Delta}\)

\(S_0S_1...S_k[n]=AB_0B_1...B_n[n]\)

評価

(0)(1)(2)(2)=|(0,0)(1,0)(2,0)|(1,1)(2,0)(3,0)|(2,2)(3,0)(4,0)|...=B(0,0,0)(1,1,1)

(0)(1)(2)(3)(3)=(0,0)(1,0)(2,0)(3,0)|(2,2)(3,0)(4,0)(5,0)|(4,4)(5,0)(6,0)(7,0)|...

(0,0)(1,0)(2,0)(3,0)(2,2)=(0,0,0)(1,0,0)(2,0,0)(3,0,0)|(2,1,1)(3,0,0)(4,0,0)(5,0,0)|(4,2,2)(5,0,0)(6,0,0)(7,0,0)|...

(0)(1)(2)(3)(4)(4)=|(0,0)(1,0)(2,0)(3,0)(4,0)|(3,3)(4,0)(5,0)(6,0)(7,0)|(6,6)(7,0)(8,0)(9,0)(10,0)|...

(0,0)(1,0)(2,0)(3,0)(4,0)(3,3)=|(0,0,0)(1,0,0)(2,0,0)(3,0,0)(4,0,0)|(3,2,2)(4,0,0)(5,0,0)(6,0,0)(7,0,0)|...

(0,0,0)(1,0,0)(2,0,0)(3,0,0)(4,0,0)(3,2,2)=|(0,0,0,0)(1,0,0,0)(2,0,0,0)(3,0,0,0)(4,0,0,0)|(3,2,1,1)(4,0,0,0)(5,0,0,0)(6,0,0,0)(7,0,0,0)|...

(0,0)(1,1)=B(0(0)0)(1(0)1)

(0,0)(1,1)(1,1)=B(0,0(0)0)(1,1(0)1)(1,1(0)0)

(0,0)(1,1)(2,0)=(0,0)|(1,1)|(1,1)|(1,1)|...=B(0,0(0)0)(1,1(0)1)(1,1(0)0)(2,0(0)0)

(0,0)(1,1)(2,0)(2,0)=|(0,0)(1,1)(2,0)|(1,1)(2,2)(3,0)|(2,2)(3,3)(4,0)|...

(0,0)(1,1)(2,0)(1,1)(2,2)=(0,0)(1,1)(2,0)|(1,1)|(2,1)|(3,1)|...

(0,0)(1,1)(2,0)(1,1)(2,1)=|(0,0)(1,1)(2,0)(1,1)|(2,0)(3,1)(4,0)(3,1)|(4,0)(5,1)(6,0)(5,1)|...

(0,0)(1,1)(2,0)(1,1)(2,0)(3,1)=B(0,0,0(0)0)(1,1,1(0)1)(1,1,0(0)0)(2,2,1(0)1)

(0,0)(1,1)(2,0)(1,1)(2,0)(3,1)(2,1)=B(0,0,0(0)0)(1,1,1(0)1)(1,1,0(0)0)(2,2,1(0)1)(2,1,0(0)0)

(0,0)(1,1)(2,0)(1,1)(2,0)(3,1)(2,2)=B(0,0,0(0)0)(1,1,1(0)1)(1,1,0(0)0)(2,2,1(0)1)(2,2,0(0)0)

(0,0)(1,1)(2,0)(1,1)(2,0)(3,1)(3,0)=B(0,0,0(0)0)(1,1,1(0)1)(1,1,1(0)0)

(0,0)(1,1)(2,0)(1,1)(2,0)(3,1)(3,1)=B(0,0,0(0)0)(1,1,1(0)1)(1,1,1(0)0)(2,1,0(0)0)

(0,0)(1,1)(2,0)(1,1)(2,0)(3,1)(4,0)=B(0,0,0(0)0)(1,1,1(0)1)(1,1,1(0)0)(2,1,0(0)0)(2,0,0(0)0)

(0,0)(1,1)(2,0)(1,1)(2,0)(3,1)(4,0)(3,1)(4,0)(5,0)(4,1)=B(0,0,0(0)0)(1,1,1(0)1)(1,1,1(0)0)(2,1,0(0)0)(2,1,0(0)0)

(0,0)(1,1)(2,0)(1,1)(2,0)(3,1)(4,0)(3,1)(4,0)(5,1)=B(0,0,0(0)0)(1,1,1(0)1)(1,1,1(0)0)(2,1,0(0))(3,2,1(0)1)

(0,0)(1,1)(2,0)(1,1)(2,1)=B(0,0,0(0)0)(1,1,1(0)1)(1,1,1(0)0)(2,1,0(0)0)(3,2,1(0)1)(3,2,1(0)0)(4,2,0(0)0)

超限行列システム(TMs)

表記

• \(k,m,n:\)自然数
• \(a_{x,y}:\)非負整数
• \(S_x=(a_{x,0},a_{x,1},...)\)

• 正規形\(:S_0,S_1,...S_k[n]\)

定義

• \(Z:\)零ベクトル
• \(b,c,e,x,y:\)非負整数

rule1:\(Z[n]=n+1\)

rule2:\(S_0S_1...S_{k-1}Z[n]=S_0S_1...S_{k-1}[n+1]\)

rule3:

• \(dim(x)=Max\{b|a_{x,b}>0\}\)
• \(p_0(x)=Max\{b|(b<x)∧(a_{b,0}+1=a_{x,0})\}\)
• \(p_{y+1}(x)=Max\{b|(b<x)∧(a_{b,y+1}+1=a_{x,y+1})∧(∃c[p_y^c(x)=b])\}\)
• \(P_0(x)=Max\{b|(b<x)∧(a_{b,0}<a_{x,0})∧((P_0(b)≠b-1)∨(a_{b,0}=0))\}\)
• \(P_{y+1}(x)=Max\{b|(b<x)∧(a_{b,y+1}<a_{x,y+1})∧\)

\(((P_{y+1}(b)≠b-1)∨(a_{b,y+1}=0))∧(∃c[P_y^c(x)=b])\}\)

• \(r=\begin{cases}Max\{b|∀y,y≤dim(k)[∃c[p_y^c(k)=b]]\}&\text{if}　a_{k-1,0}+1=a_{k,0}\\Max\{b|∀y,y≤dim(k)[∃c[P_y^c(k)=b]]\}&\text{otherwise}\end{cases}\)
• \({\Delta}=D_0D_1...D_{k-1-r}\)

• \(D_x=(d_{x,0},d_{x,1},...d_{x,m})\)

• \(d_{0,y}=\begin{cases}a_{k,y}-a_{r,y}&\text{if}　y<dim(k)\\a_{k,dim(k)}-a_{r,dim(k)}-1&\text{if}　dim(k)≤y≤dim(k)+n\\0&\text{otherwise}\end{cases}\)
• \(d_{x+1,y}=\begin{cases}d_{0,y}&\text{if}　∃b[P_y^b(r+x+1)=r]\\0&\text{otherwise}\end{cases}\)

• \(A=S_0S_1...S_{r-1}\)
• \(B_0=S_rS_{r+1}...S_{k-1}\)
• \(B_b=B_0+b×{\Delta}\)

\(S_0S_1...S_k[n]=AB_0B_1...B_n[n]\)

評価

(0)(1)(2)(2)=|(0)(1)(2)|(1,1,1...1)(2)(3)|(2,2,2...2)(3)(4)|...=B(0(0)0)(1(0)1)

(0)(1)(2)(2)(2)=|(0)(1)(2)(2)|(1,1,1...1)(2)(3)(3)|(2,2,2...2)(3)(4)(4)|...=B(0(0)0)(1(0)1)(1(0)1)

(0)(1)(2)(3)=B(0(0)0)(1(0)1)(2(0)0)

(0)(1)(2)(3)(2,1)=B(0,0(0)0)(1,1(0)1)(2,1(0)0)

(0)(1)(2)(3)(2,1)(2)=|(0)(1)(2)(3)(2,1)|(1,1,1...1)(2)(3)(4)(3,2)|(2,2,2...2)(3)(4)(5)(4,3)|...

(0)(1)(2)(3)(2,1)(1,1)(2)(3)(4)(3,2)=(0)(1)(2)(3)(2,1)|(1,1)(2)(3)(4)|(3,1)(4)(5)(6)|(4,1)(5)(6)(7)|...

(0)(1)(2)(3)(2,1)(1,1)(2)(3)(4)(3,1)(3)=(0)(1)(2)(3)(2,1)|(1,1)(2)(3)(4)(3,1)|(2,2,1...1)(3)(4)(5)(4,1)|(3,3,2...2)(4)(5)(6)(5,1)|...=B(0,0,0(0)0)(1,1,1(0)1)(2,1,0(0)0)(1,1,0(0)0)(2,2,1(0)1)(3,1,0(0)0)(2,2,1(0)1)

(0)(1)(2)(3)(2,1)(1,1)(2)(3)(4)(3,2)=B(0,0,0(0)0)(1,1,1(0)1)(2,1,0(0)0)(1,1,0(0)0)(2,2,1(0)0)(3,2,0(0)0)

(0)(1)(2)(3)(2,1)(1,1,1)=B(0,0,0(0)0)(1,1,1(0)1)(2,1,0(0)0)(1,1,1(0)0)

(0)(1)(2)(3)(2,1)(2)=B(0,0(0)0)(1,1(0)1)(2,1(0)0)(1,1(0)1)

(0)(1)(2)(3)(2,1)(2,1)=B(0,0(0)0)(1,1(0)0)(2,1(0)0)(1,1(0)1)(2,1(0)0)

(0)(1)(2)(3)(2,1)(3)=B(0,0(0)0)(1,1(0)0)(2,1(0)0)(2,0(0)0)

(0)(1)(2)(3)(2,1)(3,1)=B(0,0(0)0)(1,1(0)0)(2,1(0)0)(2,1(0)0)

(0)(1)(2)(3)(2,2)=|(0)(1)(2)(3)|(2,1,1...1)(3)(4)(5)|(4,2,2...2)(5)(6)(7)|...=B(0,0(0)0)(1,1(0)0)(2,1(0)1)

(0)(1)(2)(3)(2,2,2)=B(0,0,0(0)0)(1,1,1(0)1)(2,2,1(0)1)

(0)(1)(2)(3)(3)=B(0(0)0)(1(0)1)(2(0)1)

(0)(1,1)=B(0(0)0)(1(0)1)(2(0)2)

(0)(1,1)(1,1)=B(0,0(0)0)(1,1(0)1)(2,2(0)2)(1,1(0)0)

(0)(1,1)(2)=B(0,0(0)0)(1,1(0)1)(2,2(0)2)(1,1(0)0)(2,0(0)0)

(0)(1,1)(2)(2)=|(0)(1,1)(2)|(1,1,1...1)(2,2)(3)|(2,2,2...2)(3,3)(4)|...

(0)(1,1)(2)(1,1,1)=|(0)(1,1)(2)|(1,1)(2,2)(3)|(2,2)(3,3)(4)|...

(0)(1,1)(2)(1,1)(2,2)=(0)(1,1)(2)|(1,1)|(2,1)|(3,1)|...

(0)(1,1)(2)(1,1)(2,1)=|(0)(1,1)(2)(1,1)|(2)(3,1)(4)(3,1)|(4)(5,1)(6)(5,1)|...

(0)(1,1)(2)(1,1)(2)(3)=B(0,0(0)0)(1,1(0)1)(2,2(0)2)(1,1(0)0)(2,0(0)0)(3,0(0)0)

(0)(1,1)(2)(1,1)(2)(3)(3)=(0)(1,1)(2)|(1,1)(2)(3)|(2,2,1...1)(3)(4)|(3,3,2...2)(4)(5)|...=B(0,0,0(0)0)(1,1,1(0)1)(2,2,2(0)2)(1,1,0(0)0)(2,2,1(0)1)

(0)(1,1)(2)(1,1)(2)(3,1)=B(0,0,0(0)0)(1,1,1(0)1)(2,2,2(0)2)(1,1,0(0)0)(2,2,1(0)1)(3,3,2(0)2)

(0)(1,1)(2)(1,1)(2)(3,1)(3)=(0)(1,1)(2)|(1,1)(2)(3,1)|(2,2,1...1)(3)(4,2,1...1)|(3,3,2...2)(4)(5,3,2...2)|...=B(0(0)0)(1(0)1)(2(0)2)(1(0)1)

(0)(1,1)(2)(1,1)(2)(3,1)(3,1)=B(0,0(0)0)(1,1(0)1)(2,2(0)2)(1,1(0)1)(2,1(0)0)

(0)(1,1)(2)(1,1)(2)(3,1)(4)=B(0,0(0)0)(1,1(0)1)(2,2(0)2)(1,1(0)1)(2,1(0)0)(3,0(0)0)

(0)(1,1)(2)(1,1)(2)(3,1)(4)(3,1)(4)(5,1)=B(0,0,0(0)0)(2,2,2(0)2)(1,1,1(0)1)(2,1,0(0)0)(3,2,1(0)1)(4,3,2(0)2)

(0)(1,1)(2)(1,1)(2,1)=B(0,0,0(0)0)(1,1,1(0)1)(2,2,2(0)2)(1,1,1(0)1)(2,1,1(0)0)

第2大偽原始数列(SLPpSs)

\(LSZNOS+NB\)

表記

\((a_0,a_1,a_2...a_k)[n]\)

サブルール

• 大小関係

\(A<B⇔∃C[((A<C)∨(A=C))∧(∃c[(\text{ex}(B,c)=C)])]\)

※\(A、B、C\)は列

• 展開

列\(S=(s_0,s_1,s_2...s_m)\)と自然数\(b\)に対して\(\text{ex}(S,b)\)を以下の通り定義する。

• \(r=Max\{c|(s_{c-1}≥s_c)∧(s_c<s_m)\}\)
• \(q(0)=\begin{cases}Max\{c|s_{c-1}=s_r-1\}&\text{if}　∃c\\0&\text{otherwise}\end{cases}\)
• \(q(c+1)=min\{d|(s_d=s_r)∧(q(c)<d)\}\)
• \(l=Max\{c|q(c)=r\}\)
• \(S_c=(s_{q(c)},s_{q(c)+1},s_{q(c)+2}...s_{q(c+1)-1})\)
• \(S_l=(s_{q(l)},s_{q(l)+1},s_{q(l)+2}...s_m)\)
• \(i=\begin{cases}Max\{q(c)|S_c<S_l\}&\text{if}　∃q(c)\\q(0)&\text{otherwise}\end{cases}\)
• \(A=(s_0,s_1,s_2...s_{i-1})\)
• \({\Delta}=s_m-s_i-1\)
• \(B_c=(s_i,s_{i+1},s_{i+2}...s_{m-1})+{\Delta}×c\)
• \(\text{ex}(S,b)=A\frown B_0\frown B_1\frown B_2...B_b\)

計算法

rule1:\((0)[n]=n+1\)

rule2:\((a_0,a_1,a_2...a_{k-1},0)[n]=(a_0,a_1,a_2...a_{k-1})[n+1]\)

rule3:\(\text{ex}((a_0,a_1,a_2...a_k),n)[n]\)

評価

(0,1,2)=\({\beta_0}({\beta_0}({\Omega}))\)

(0,1,2,0,1,1,2,3)=\({\beta_0}({\beta_0}({\Omega}),0,{\beta_0}({\Omega}))\)

(0,1,2,0,1,1,2,3,0,1,2)=(|0,1,2,0,1,1,2,3|1,2,3,1,2,2,3,4|2,3,4,2,3,3,4,5|...)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega})))\)

(0,1,2,0,1,1,2,3,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega})))\)

(0,1,2,0,1,1,2,3,1,0,1,1,2,3,0,1,2)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega}),1,0,{\beta_0}({\Omega})))\)

(0,1,2,0,1,1,2,3,1,0,1,1,2,3,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega}),1,1))\)

(0,1,2,0,1,1,2,3,1,0,1,2)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},1,{\beta_0}({\Omega}))))\)

(0,1,2,0,1,1,2,3,1,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega}),2))\)

(0,1,2,0,1,1,2,3,1,1,2)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega}),{\omega}))\)

(0,1,2,0,1,1,2,3,1,1,2,3)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega}),{\beta_0}({\beta_0}({\Omega}))))\)

(0,1,2,0,1,1,2,3,1,2)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega}),{\beta_0}({\beta_0}({\Omega}),0)))\)

(0,1,2,0,1,2)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega}),{\beta_0}({\Omega})))\)

(0,1,2,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0)))\)

(0,1,2,2)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega}))))\)

(0,1,2,2,0,1,2,2)=(|0,1,2,2,0,1,2|1,2,3,3,1,2,3|2,3,4,4,2,3,4|...)

(0,1,2,2,0,1,2)=(|0,1,2,2,0,1|1,2,3,3,1,2|2,3,4,4,2,3|...)

(0,1,2,2,0,1,1,2,3,0,1,2)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),0,{\beta_0}({\Omega})))\)

(0,1,2,2,0,1,1,2,3,0,1,2,0,1,1,2,3,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),0,{\beta_0}({\Omega}),1))\)

(0,1,2,2,0,1,1,2,3,0,1,2,0,1,2)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),0,{\beta_0}({\Omega}),{\beta_0}({\Omega})))\)

(0,1,2,2,0,1,1,2,3,0,1,2,0,1,2,2)=(0,1,2,2,0,1,1,2,3,0,1,2|0,1,2|1,2,3|2,3,4|...)

=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),0,{\beta_0}({\Omega},0,{\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})))))))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),0,{\beta_0}({\Omega},0,{\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega}))))),1))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,0,1,1,2,3,0,1,2,1)

=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),0,{\beta_0}({\Omega},0,{\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega}))))),1,1))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,0,1,1,2,3,1)

=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),0,{\beta_0}({\Omega},0,{\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega}))))),1,2))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,0,1,2)

=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),0,{\beta_0}({\Omega},0,{\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega}))))),{\beta_0}({\Omega})))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,0,1,2,1,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),0,{\beta_0}({\Omega},0,{\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})))),0)))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,0,1,2,2)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),0,{\beta_0}({\Omega},0,{\beta_0}({\Omega}))))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),1))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,1,0,1,1,2,3,0,1,2,1,2,3,3,0,1,2,2)

=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),1,0,{\beta_0}({\Omega},0,{\beta_0}({\Omega}))))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,1,0,1,1,2,3,0,1,2,1,2,3,3,0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,1)

=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),1,0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),1))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,1,0,1,1,2,3,0,1,2,1,2,3,3,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),1,1))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,1,0,1,1,2,3,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),1,2))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,1,0,1,2)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),1,{\beta_0}({\Omega})))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,1,0,1,2,0,1,2,2)

=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),1,{\beta_0}({\Omega},0,{\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})))))))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,1,0,1,2,1,2,3,3,0,1,2,2)

=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),1,{\beta_0}({\Omega},0,{\beta_0}({\Omega}))))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,1,0,1,2,1,2,3,3,0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,0,1,2,2)

=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),1,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),0,{\beta_0}({\Omega},0,{\beta_0}({\Omega}))))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,1,0,1,2,1,2,3,3,0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,1)

=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),1,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),1))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,1,0,1,2,1,2,3,3,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),1,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),1))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,1,0,1,2,1,2,3,3,1,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),2))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,1,0,1,2,1,2,3,3,1,1,0,1,1,2,3,0,1,2)

=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),2,0,{\beta_0}({\Omega})))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,1,0,1,2,1,2,3,3,1,1,0,1,1,2,3,0,1,2,1,2,3,3,0,1,2,2)

=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),2,0,{\beta_0}({\Omega},0,{\beta_0}({\Omega}))))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,1,0,1,2,1,2,3,3,1,1,0,1,1,2,3,0,1,2,1,2,3,3,0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,1)

=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),2,0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),1))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,1,0,1,2,1,2,3,3,1,1,0,1,1,2,3,0,1,2,1,2,3,3,1)

=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),2,1))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,1,0,1,2,1,2,3,3,1,1,0,1,1,2,3,0,1,2,1,2,3,3,1,0,1,1,2,3,1)

=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),2,1,2))\)

(0,1,2,2,0,1,1,2,3,0,1,2,1,2,3,3,1,0,1,2,1,2,3,3,1,1,0,1,1,2,3,0,1,2,1,2,3,3,1,0,1,2,1,2,3,3,0,1,2,2)

=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega})),2,1,{\beta_0}({\Omega},0,{\beta_0}({\Omega}))))\)

第3大偽原始数列(TLPpSs)

\(LSZNOS+NB\)

表記

\((a_0,a_1,a_2...a_k)[n]\)

サブルール

• 大小関係

\(A<B⇔∃C[((A<C)∨(A=C))∧(∃c[(\text{ex}(B,c)=C)])]\)

※\(A、B、C\)は列

• 展開

列\(S=(s_0,s_1,s_2...s_m)\)と自然数\(b\)に対して\(\text{ex}(S,b)\)を以下の通り定義する。

• \(p(c,d)=\begin{cases}Max\{e|(s_e=d)∧(e≤c)\}&\text{if}　∃e\\0&\text{otherwise}\end{cases}\)
• \(q(c,0)=\begin{cases}Max\{e|s_{e-1}=c-1\}&\text{if}　∃e\\0&\text{otherwise}\end{cases}\)
• \(q(c,d+1)=min\{e|(q(c,d)<e)∧(s_e=c)\}\)
• \(S_{c,d}=(s_{q(c,d)},s_{q(c,d)+1},s_{q(c,d)+2}...s_{q(c,d+1)-1})\)
• \(l_c=Max\{d|q(c,d)=p(m,c)\}\)
• \(S_{c,l_c}=(s_{p(m,c)},s_{p(m,c)+1},s_{p(m,c)+2}...s_m)\)
• \(i=\begin{cases}Max\{q(c,d)|S_{c,d}<S_{c,l_c}\}&\text{if}　∃q(c,d)\\0&\text{otherwise}\end{cases}\)
• \(A=(s_0,s_1,s_2...s_{i-1})\)
• \({\Delta}=s_m-s_i-1\)
• \(B_c=(s_i,s_{i+1},s_{i+2}...s_{m-1})+{\Delta}×c\)
• \(\text{ex}(S,b)=A\frown B_0\frown B_1\frown B_2...B_b\)

計算法

rule1:\((0)[n]=n+1\)

rule2:\((a_0,a_1,a_2...a_{k-1},0)[n]=(a_0,a_1,a_2...a_{k-1})[n+1]\)

rule3:\(\text{ex}((a_0,a_1,a_2...a_k),n)[n]\)

\(m\)-階差数列システム

​表記

\((a_0,a_1...a_k)_m[n]\)

​計算法

rule1:\((a_0,a_1...a_{k-1},0)_m[n]=(a_0,a_1...a_{k-1})_m[n+1]\)

rule2:

• \(D=max\{a_k-a_{k-1},-m\}\)
• \(i=Max\{j|(a_j-a_{j-1}<D)∧(a_j<a_k)\}\)
• \({\Delta}=a_k-a_i-1\)
• \(A=(a_0,a_1,...a_{i-1})_m\)
• \(B_p=(a_i+{\Delta}×p,a_{i+1}+{\Delta}×p,...a_{k-1}+{\Delta}×p)_m\)
• \((a_0,a_1...a_k)_m[n]=\{A\frown B_0\frown B_1\frown...B_n\}[n+1]\)

評価

• \(m=-2→\)原始数列

(0,1,2,...)=(0,1,3)=\({\varepsilon_0}\)

(0,1,3,0,1,3)=\({\varepsilon_0}×2\)

(0,1,3,1)=\({\varepsilon_0}×{\omega}\)

(0,1,3,1,3)=\({\varepsilon_0}^2\)

(0,1,3,2)=\({\varepsilon_0^{\omega}}\)

(0,1,3,2,4)=\({\varepsilon_0^{\varepsilon_0}}\)

(0,1,3,3)=\({\varepsilon_1}\)

(0,1,3,4)=\({\psi_0({\omega})}\)

(0,1,3,4,6)=\({\psi_0({\psi_0(0)})}\)

(0,1,3,5)=(0|1,3|4,6|7,9|...)=\({\psi_0({\Omega})}\)

(0,1,3,5,3)=\({\psi_0({\Omega}+1)}\)

(0,1,3,5,3,5)=\({\psi_0({\Omega+{\psi_0(0)}})}\)

(0,1,3,5,4)=\({\psi_0({\Omega+{\psi_0(0)×{\omega}}})}\)

(0,1,3,5,5)=\({\psi_0({\Omega+{\psi_0(1)}})}\)

(0,1,3,5,6)=\({\psi_0({\Omega+{\psi_0({\omega})}})}\)

(0,1,3,5,6,8,10)=\({\psi_0({\Omega+{\psi_0({\Omega})}})}\)

(0,1,3,5,7)=\({\psi_0({\Omega×2})}\)

(0,1,3,6)=\({\psi_0({\Omega×{\omega}})}\)

(0,1,3,6,8,11)=\({\psi_0({\Omega×{\psi_0({\Omega})}})}\)

(0,1,3,6,9)=\({\psi_0({\Omega^2})}\)

(0,1,3,6,10)=\({\psi_0({\Omega^2×{\omega}})}\)

(0,1,3,6,10...)=\({\psi_0({\Omega^{\omega}})}\)

• \(m=-1→\)大偽原始数列
• \(m=0\)

(0,1,2)=(|0,1|1,2|2,3|...)

(0,1,1,2,2)=(|0,1,1,2|1,2,2,3|2,3,3,4|...)=\({\varepsilon_0}\)

(0,1,1,2,2,3)=\({\psi_0({\omega})}\)

(0,1,1,2,2,3,2,3,3,4,4)=\({\psi_0({\varepsilon_0})}\)

(0,1,1,2,2,3,3)=\({\psi_0({\Omega})}\)

(0,1,2)=\({\psi_0({\Omega^{\omega}})}\)

(0,1,2,2)=\({\psi_0({\Omega^{\omega}}+1)}\)

(0,1,2,2,3,3)=\({\psi_0({\Omega^{\omega}+{\Omega}})}\)

(0,1,2,2,3,4)=\({\psi_0({\Omega^{\omega}×2})}\)

(0,1,2,3)=\({\psi_0({\Omega^{\omega}×{\omega}})}\)

(0,1,2,3,3,4,5,6)=\({\psi_0({\Omega^{\omega}×{\omega×2}})}\)

(0,1,2,3,4)=\({\psi_0({\Omega^{\omega}×{\omega^2}})}\)

(0,1,3)=\({\psi_0({\Omega^{\omega}×{\omega^{\omega}}})}\)

(0,1,3,3,4,6)=\({\psi_0({\Omega^{\omega}×{\omega^{\omega}×2}})}\)

(0,1,3,4)=\({\psi_0({\Omega^{\omega}×{\omega^{\omega+1}}})}\)

(0,1,3,4,6)=\({\psi_0({\Omega^{\omega}×{\omega^{\omega×2}}})}\)

(0,1,3,5)=\({\psi_0({\Omega^{\omega}×{\omega^{\omega^2}}})}\)

(0,1,3,6)=\({\psi_0({\Omega^{\omega}×{\omega^{\omega^{\omega}}}})}\)

(0,1,3,6,10...)=\({\psi_0({\Omega^{\omega}×{\varepsilon_0}})}\)

\(L-\)階差数列システム

[1]

\(m\)-ヒドラ数列システム

​表記

\((a_0,a_1...a_k)_m[n]\)

​計算法

rule1:\((a_0,a_1...a_{k-1},0)_m[n]=(a_0,a_1...a_{k-1})_m[n+1]\)

rule2:

• \(p(b)=Max\{c|a_c<a_b+m+1\}\)
• \(i=\begin{cases}Max\{b|(a_b<a_k)∧(∃c[p^c(k)=b])\}&\text{if}　a_k+m-a_{p(k)}=0\\Max\{b|(a_b<a_k)∧(∃c[p^c(k)=b])∧(a_b-a_{p(b)}<a_k-a_{p(k)})\}&\text{otherwise}\end{cases}\)
• \({\Delta}=a_k-a_i-1\)
• \(A=(a_0,a_1...a_{i-1})_m\)
• \(B_b=(a_i+{\Delta}×b,a_{i+1}+{\Delta}×b,...a_{k-1}+{\Delta}×b)_m\)
• \((a_0,a_1...a_k)_m[n]=A\frown B_0\frown B_1\frown...B_n[n]\)

評価

• m=-1→原始数列

(0,1,3)=(0|1|2|3|...)=\({\varepsilon_0}\)

(0,1,3,1,3)=(0,1,3|1|2|3|...)=\({\varepsilon_0^2}\)

(0,1,3,2)=(0|1,3|1,3|1,3|...)=\({\varepsilon_0^{\omega}}\)

(0,1,3,3)=(0|1,3|2,4|3,5|...)=\({\varepsilon_1}\)

(0,1,3,4)=(0,1|3|3|3|...)=\({\psi_0({\omega})}\)

(0,1,3,4,6)=(0,1,3|4|5|6|...)=\({\psi_0({\psi_0(0)})}\)

(0,1,3,5)=(0|1,3|4,6|7,9|...)=\({\psi_0({\Omega})}\)

(0,1,3,5,3)=(0|1,3,5|2,4,6|3,5,7|...)=\({\psi_0({\Omega+1})}\)

(0,1,3,5,3,4,6,8)=\({\psi_0({\Omega+{\psi_0({\Omega})}})}\)

(0,1,3,5,3,5)=(0|1,3,5,3|4,6,8,6|7,9,11,9|...)=\({\psi_0({\Omega×2})}\)

(0,1,3,5,4)=\({\psi_0({\Omega×{\omega}})}\)

(0,1,3,5,5)=\({\psi_0({\Omega^2})}\)

(0,1,3,5,6)=\({\psi_0({\Omega^{\omega}})}\)

(0,1,3,5,7)=\({\psi_0({\Omega^{\Omega}})}\)

(0,1,3,5,7,9)=\({\psi_0({\Omega^{\Omega^{\Omega}}})}\)

(0,1,3,6)=\({\psi_0({\psi_1(0)})}\)

(0,1,3,6,10...)=\({\psi_0({\Omega_{\omega}})}\)

• m=0

(0,1,2)=(|0,1|1,2|2,3|...)=\({\varepsilon_0}\)

(0,1,2,2)=(0|1,2|1,2|1,2|...)=\({\varepsilon_0×{\omega^{\omega^2}}}\)

(0,1,2,2,3,4)=\({\varepsilon_0^2}\)

(0,1,2,3)=(|0,1,2|2,3,4|4,5,6|...)=\({\varepsilon_1}\)

(0,1,3)=(0|1|2|3|...)=\({\psi_0({\omega})}\)

(0,1,3,2)=\({\psi_0({\omega}+1)}\)

(0,1,3,2,3)=\({\psi_0({\omega+2})}\)

(0,1,3,2,4)=\({\psi_0({\omega×2})}\)

(0,1,3,3)=(0|1,3|2,4|3,5|...)=\({\psi_0({\omega^2})}\)

(0,1,3,3,2,4,4)=\({\psi_0({\omega^2×2})}\)

(0,1,3,3,4)=\({\psi_0({\omega^{\omega}})}\)

(0,1,3,3,4,5)=\({\psi_0({\psi_0(0)})}\)

(0,1,3,4)=\({\psi_0({\Omega})}\)

(0,1,3,4,2)=\({\psi_0({\Omega+1})}\)

(0,1,3,4,2,4,5)=\({\psi_0({\Omega×2})}\)

(0,1,3,4,3)=\({\psi_0({\Omega×{\omega}})}\)

(0,1,3,4,3,2,4,5)=\({\psi_0({\Omega×({\omega+1})})}\)

(0,1,3,4,3,3)=\({\psi_0({\Omega×{\omega×2}})}\)

(0,1,3,4,3,4)=\({\psi_0({\Omega×{\omega^2}})}\)

(0,1,3,4,4)=\({\psi_0({\Omega×{\omega^{\omega}}})}\)

(0,1,3,4,4,5,7,8)=\({\psi_0({\Omega×{\psi_0({\Omega})}})}\)

(0,1,3,4,5)=\({\psi_0({\Omega^2})}\)

(0,1,3,4,5,2,4,5,6)=\({\psi_0({\Omega^2×2})}\)

(0,1,3,4,6)=\({\psi_0({\Omega^{\omega}})}\)

(0,1,3,4,6,3)=\({\psi_0({\Omega^{\omega}×2})}\)

(0,1,3,4,6,5)=\({\psi_0({\Omega^{\omega+1}})}\)

(0,1,3,4,6,5,7)=\({\psi_0({\Omega^{\omega×2}})}\)

(0,1,3,4,6,6)=\({\psi_0({\Omega^{\omega^{\omega}}})}\)

(0,1,3,4,6,7)=\({\psi_0({\Omega^{\Omega}})}\)

(0,1,3,5)=(0|1,3|4,6|7,9|...)=\({\psi_0({\psi_1(0)})}\)

(0,1,3,5,3)=\({\psi_0({\psi_1(0)×{\omega}})}\)

(0,1,3,5,4)=\({\psi_0({\psi_1(0)×{\Omega}})}\)

(0,1,3,5,5)=\({\psi_0({\psi_1(0)^{\omega}})}\)

(0,1,3,5,6)=\({\psi_0({\psi_1(0)^{\Omega}})}\)

(0,1,3,5,7)=\({\psi_0({\psi_1(1)})}\)

(0,1,3,5,7,9)=\({\psi_0({\psi_1(2)})}\)

(0,1,3,6)=\({\psi_0({\psi_1({\omega})})}\)

(0,1,3,6,10...)=\({\psi_0({\Omega_{\omega}})}\)

• m=1

(0,1,1,2,2)=(|0,1,1,2|1,2,2,3|2,3,3,4|...)=\({\varepsilon_0}\)

(0,1,1,2,2,2)=(|0,1,1,2,2|1,2,2,3,3|2,3,3,4,4|...)=\({\varepsilon_1}\)

(0,1,1,2,2,3)=\({\psi_0({\omega})}\)

(0,1,1,2,2,3,2)=(0,1|1,2,2,3|1,2,2,3|1,2,2,3|...)=\({\psi_0({\omega})×{\omega^{\omega^{\omega+1}}}}\)

(0,1,1,2,2,3,2,2)=(|0,1,1,2,2,3,2|1,2,2,3,3,4,3|2,3,3,4,4,5,4|...)=\({\psi_0({\omega+1})}\)

(0,1,1,2,2,3,2,3)=\({\psi_0({\omega×2})}\)

(0,1,1,2,2,3,2,3,3)=\({\psi_0({\omega^2})}\)

(0,1,1,2,2,3,3)=(|0,1,1,2,2,3|2,3,3,4,4,5|4,5,5,6,6,7|...)=\({\psi_0({\Omega})}\)

(0,1,1,2,2,3,3,2,2)=\({\psi_0({\Omega+1})}\)

(0,1,1,2,2,3,3,2,3)=\({\psi_0({\Omega+{\omega}})}\)

(0,1,1,2,2,3,3,3)=\({\psi_0({\Omega×2})}\)

(0,1,1,2,2,3,3,4)=\({\psi_0({\Omega×{\omega}})}\)

(0,1,1,2,2,3,3,4,3,3)=\({\psi_0({\Omega×({\omega+1})})}\)

(0,1,1,2,2,3,3,4,4)=\({\psi_0({\Omega^2})}\)

(0,1,2)=\({\psi_0({\Omega^{\omega}})}\)

第2-\(m\)-階差数列

​表記

\((a_0,a_1,・・・a_k)_m[n]\)

​計算法

rule1:\((a_0,a_1,・・・a_{k-1},0)_m[n]=(a_0,a_1,・・・a_{k-1})_m[n+1]\)

rule2:

• \(b=Max\{-m,a_k-a_{k-1}\}\)
• \(c_l=\begin{cases}0&\text{if}　l=0\\min\{d|(a_d-a_{d-1}<b)∧(c_{l-1}<d)\}&\text{otherwise}\end{cases}\)
• \(p(d)=Max\{e|(a_e<a_d)∧(e<d)∧(∃f[e=c_f])\}\)
• \(i=\begin{cases}p(k)&\text{if}　a_k-a_{p(k)}=1\\Max\{d|(∃e[p^e(k)=d])∧(a_k-a_{p(k)}>a_d-a_{p(d)})\}&\text{otherwise}\end{cases}\)
• \(A=(a_0,a_1...a_{i-1})_m\)
• \(B_l=(a_i,a_{i+1}...a_{k-1})_m+(a_k-a_i-1)×l\)
• \((a_0,a_1,・・・a_{k-1},a_k)_m[n]=A\frown B_0\frown B_1\frown...B_k[n]\)

補足

• \(b\)を最小値とする階差表記に変換→先頭の項で-1-ヒドラ数列を計算
• \(p(d)\)は階差表記の先頭部分

例

\((0,1,2,3,3,4,5,5,6,7,8)_{-1}→(0[1,1,1],3[1,1],5[1,1],8)→(0,3,5,8)→\)

\((0,3|5|7||9|...)→(0,1,2,3,3,4,5|5,6,7|7,8,9|9,10,11|...)\)

評価

• m=-1

(0,1,2)=(0[1],2)=\({\varepsilon_0}\)

(0,1,2,2)=(0[1,1],2)=\({\varepsilon_1}\)

(0,1,2,2,3)=(0[1,1],2,3)=(0,1,2|2|2|2|...)=\({\psi_0({\omega})}\)

(0,1,2,2,3,4)=(0[1,1],2[1],4)=(|0,1,2,2,3|3,4,5,5,6|6,7,8,8,9|...)=\({\psi_0({\Omega})}\)

(0,1,2,2,3,4,2,3,4)=(0[1,1],2[1,1],2[1],4)=\({\psi_0({\Omega×2})}\)

(0,1,2,2,3,4,3)=\({\psi_0({\Omega×{\omega}})}\)

(0,1,2,2,3,4,4)=\({\psi_0({\Omega^2})}\)

(0,1,2,2,3,4,4,5)=\({\psi_0({\Omega^{\omega}})}\)

(0,1,2,2,3,4,4,5,6)=\({\psi_0({\Omega^{\Omega}})}\)

(0,1,2,3)=\({\psi_0({\psi_1(0)})}\)

(0,1,2,3,2)=\({\psi_0({\psi_1(0)+1})}\)

(0,1,2,3,2,3,4)=\({\psi_0({\psi_1(0)+{\Omega}})}\)

(0,1,2,3,2,3,4)=\({\psi_0({\psi_1(0)×2})}\)

(0,1,2,3,2,3,4,4)=\({\psi_0({\psi_1(0)×{\Omega}})}\)

(0,1,2,3,2,3,4,5)=\({\psi_0({\psi_1(0)^2})}\)

(0,1,2,3,2,3,4,5,4,5,6,7)=\({\psi_0({\psi_1(0)^{\psi_1(0)}})}\)

(0,1,2,3,3)=\({\psi_0({\psi_1(1)})}\)

(0,1,2,3,3,4,5)=\({\psi_0({\psi_1({\Omega})})}\)

(0,1,2,3,3,4,5,5,6,7,8)=\({\psi_0({\psi_1({\psi_1(0)})})}\)

(0,1,2,3,3,4,5,6)=\({\psi_0({\psi_1({\Omega_2})})}\)

(0,1,3)=\({\psi_0({\Omega_{\omega}})}\)

(0,1,3,3)=(0[1,2],3)=(|0,1,3|2,3,5|4,5,7|...)

(0,1,3,2)=\({\psi_0({\Omega_{\omega}+1})}\)

(0,1,3,2,3)=\({\psi_0({\Omega_{\omega}+{\omega}})}\)

(0,1,3,2,3,4)=\({\psi_0({\Omega_{\omega}+{\Omega}})}\)

(0,1,3,2,3,4,5)=\({\psi_0({\Omega_{\omega}+{\psi_1(0)}})}\)

(0,1,3,2,3,5)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}})}})}\)

(0,1,3,2,3,5,3)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}})×{\omega}}})}\)

(0,1,3,2,3,5,4)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}})×{\Omega}}})}\)

(0,1,3,2,3,5,4,5,6)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}})×{\Omega^2}}})}\)

(0,1,3,2,3,5,4,5,7)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}+1})^2}})}\)

(0,1,3,3)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}+1})}})}\)

(0,1,3,3,4)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}+{\omega}})}})}\)

(0,1,3,3,4,5)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}+{\Omega}})}})}\)

(0,1,3,3,4,6)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}+{\psi_1({\Omega_{\omega}})}})}})}\)

(0,1,3,4)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}+{\Omega_2}})}})}\)

(0,1,3,4,5)=(0[1,2,1],5)=(|0,1,3,4|4,5,7,8|8,9,11,12|...)

(0,1,3,4,4)=(0[1,2,1],4)=(|0,1,3,4|3,4,6,7|6,7,9,10|...)

(0,1,3,4,3,4,6,7)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}+{\Omega_2}+{\psi_1({\Omega_{\omega}+{\Omega_2}})}})}})}\)

(0,1,3,4,4)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}+{\Omega_2×2}})}})}\)

(0,1,3,4,4,5,6)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}+{\Omega_2×{\Omega}}})}})}\)

(0,1,3,4,4,5,6,6,7,9)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}+{\Omega_2×{\psi_1({\Omega_{\omega}+{\Omega_2}})}}})}})}\)

(0,1,3,4,4,5,6,7)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}+{\Omega_2^2}})}})}\)

(0,1,3,4,4,5,7)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}+{\psi_2({\Omega_{\omega}})}})}})}\)

(0,1,3,4,5)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}+{\psi_2({\Omega_{\omega}+{\Omega_3}})}})}})}\)

(0,1,3,4,6)=\({\psi_0({\Omega_{\omega}×2})}\)

(0,1,3,5)=\({\psi_0({\Omega_{\omega}×{\omega}})}\)

(0,1,3,5,6,8)=\({\psi_0({\Omega_{\omega}×({\omega+1})})}\)

(0,1,3,5,6,8,10)=\({\psi_0({\Omega_{\omega}×{\omega×2}})}\)

(0,1,3,5,7)=\({\psi_0({\Omega_{\omega}×{\omega^2}})}\)

(0,1,3,6)=\({\psi_0({\Omega_{\omega}×{\omega^{\omega}}})}\)

(0,1,3,6,10...)=\({\psi_0({\Omega_{\omega}×{\varepsilon_0}})}\)

• m=0

(0,1,1,2,2)=(0[1,0,1],2)=\({\varepsilon_0}\)

(0,1,1,2,2,3)=\({\psi_0({\omega})}\)

(0,1,1,2,2,3,2,3,3,4,4)=(0[1,0,1,0,1],2[1,0,1],4)=\({\psi_0({\Omega})}\)

(0,1,1,2,2,3,2,3,3,4,4,2,3,3,4,4)=\({\psi_0({\Omega×2})}\)

(0,1,1,2,2,3,2,3,3,4,4,3)=\({\psi_0({\Omega×{\omega}})}\)

(0,1,1,2,2,3,2,3,3,4,4,4)=\({\psi_0({\Omega^2})}\)

(0,1,1,2,2,3,2,3,3,4,4,5)=\({\psi_0({\Omega^{\omega}})}\)

(0,1,1,2,2,3,2,3,3,4,4,5,4,5,5,6,6)=\({\psi_0({\Omega^{\Omega}})}\)

(0,1,1,2,2,3,3)=\({\psi_0({\psi_1(0)})}\)

(0,1,1,2,2,3,3,2)=\({\psi_0({\psi_1(0)+1})}\)

(0,1,1,2,2,3,3,2,3,3,4,4)=\({\psi_0({\psi_1(0)+{\Omega}})}\)

(0,1,1,2,2,3,3,2,3,3,4,4,5,5)=\({\psi_0({\psi_1(0)×2})}\)

(0,1,1,2,2,3,3,2,3,3,4,4,5,5,4)=\({\psi_0({\psi_1(0)×{\Omega}})}\)

(0,1,1,2,2,3,3,2,3,3,4,4,5,5,4,5,5,6,6,7,7)=\({\psi_0({\psi_1(0)^2})}\)

(0,1,1,2,2,3,3,3)=\({\psi_0({\psi_1(1)})}\)

(0,1,1,2,2,3,3,4)=\({\psi_0({\psi_1({\omega})})}\)

(0,1,1,2,2,3,3,4,3,4,4,5,5)=\({\psi_0({\psi_1({\Omega})})}\)

(0,1,1,2,2,3,3,4,3,4,4,5,5,6,5,6,6,7,7,8,8)=\({\psi_0({\psi_1({\psi_1(0)})})}\)

(0,1,1,2,2,3,3,4,3,4,4,5,5,6,6)=\({\psi_0({\psi_1({\Omega_2})})}\)

(0,1,2)=\({\psi_0({\Omega_{\omega}})}\)

(0,1,2,2)=\({\psi_0({\Omega_{\omega}+1})}\)

(0,1,2,2,3,3)=(0[1,1,0,1],3)=(|0,1,2,2,3|2,3,4,4,5|4,5,6,6,7|...)

(0,1,2,2,3,2,3,3,4,4)=(|0[1,1,0,1],2[1,0,1],4)=\({\psi_0({\Omega_{\omega}+{\Omega}})}\)

(0,1,2,2,3,2,3,4)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}})}})}\)

(0,1,2,2,3,3)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}+1})}})}\)

(0,1,2,2,3,3,4,3,4,4,5,5)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}+{\Omega}})}})}\)

(0,1,2,2,3,3,4,4)=(|0,1,2,2,3,3,4|3,4,5,5,6,6,7|6,7,8,8,9,9,10|...)

\(={\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}+{\Omega_2}})}})}\)

(0,1,2,2,3,4)=\({\psi_0({\Omega_{\omega}×2})}\)

(0,1,2,3)=\({\psi_0({\Omega_{\omega}×{\omega^2}})}\)

(0,1,3)=\({\psi_0({\Omega_{\omega}×{\omega^{\omega}}})}\)

(0,1,3,4,6)=\({\psi_0({\Omega_{\omega}×{\omega^{\omega×2}}})}\)

(0,1,3,5)=\({\psi_0({\Omega_{\omega}×{\omega^{\omega^2}}})}\)

(0,1,3,6)=\({\psi_0({\Omega_{\omega}×{\omega^{\omega^{\omega}}}})}\)

(0,1,3,10...)=\({\psi_0({\Omega_{\omega}×{\varepsilon_0}})}\)

• m=1

(0,1,1,2,1,2,2,3,2)=(0[1,0,1,-1,1,0,1]2)=\({\varepsilon_0}\)

(0,1,1,2,1,2,2,3,2,1,2,2,3,2)=\({\varepsilon_1}\)

(0,1,1,2,1,2,2,3,2,2)=(0[1,0,1],1[1,0,1],2,2)=\({\psi_0({\omega})}\)

(0,1,1,2,1,2,2,3,2,3)=\({\psi_0({\omega^{\omega}})}\)

(0,1,1,2,1,2,2,3,2,3,2)=\({\psi_0({\omega^{\omega}+1})}\)

(0,1,1,2,1,2,2,3,2,3,2,1,2,2,3,2)=\({\psi_0({\omega^{\omega}+2})}\)

(0,1,1,2,1,2,2,3,2,3,2,2)=\({\psi_0({\omega^{\omega+1}})}\)

(0,1,1,2,1,2,2,3,2,3,3,4,3)=(0[1,0,1,-1,1,0,1,-1,1,0,1],3)

=(|0,1,1,2,1,2,2,3,2,3,3,4|2,3,3,4,3,4,4,5,4,5,5,6|4,5,5,6,5,6,6,7,6,7,7,8|...)

(0,1,1,2,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,4)=\({\psi_0({\Omega})}\)

(0,1,1,2,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,4,2)=\({\psi_0({\Omega}+1)}\)

(0,1,1,2,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,4,2,1,2,2,3,2)=\({\psi_0({\Omega+2})}\)

(0,1,1,2,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,4,2,2)=\({\psi_0({\Omega×{\omega}})}\)

(0,1,1,2,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,4,2,3,3,4,3,4,4,5,4)=\({\psi_0({\Omega^2})}\)

(0,1,1,2,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,4,3)=\({\psi_0({\Omega^{\omega}})}\)

(S_2,S_2+1,S_2+2,S_2+2,S_2+3,S_2+4,S_2+4,S_2+5,S_2+6,4)=\({\psi_0({\Omega^{\Omega}})}\)

(S_2,S_2+1,S_2+2,3)=\({\psi_0({\psi_1(0)})}\)

(0,1,1,2,2)=\({\psi_0({\Omega_{\omega}})}\)

(0,1,1,2,2,2)=(|0,1,1,2,2|1,2,2,3,3|2,3,3,4,4|...)

(0,1,1,2,2,1,2,2,3,2)=\({\psi_0({\Omega_{\omega}+1})}\)

(0,1,1,2,2,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,4)=\({\psi_0({\Omega_{\omega}+{\Omega}})}\)

(0,1,1,2,2,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,4,5,5,6,5)=\({\psi_0({\Omega_{\omega}+{\psi_1(0)}})}\)

(0,1,1,2,2,1,2,2,3,2,3,3,4,2,3,3,4,4)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}})}})}\)

(0,1,1,2,2,1,2,2,3,2,3,3,4,3)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}+1})}})}\)

(0,1,1,2,2,1,2,2,3,3)=\({\psi_0({\Omega_{\omega}×2})}\)

(0,1,1,2,2,1,2,2,3,3,2)=\({\psi_0({\Omega_{\omega}×2+1})}\)

(0,1,1,2,2,1,2,2,3,3,2,1,2,2,3,3)=\({\psi_0({\Omega_{\omega}×3})}\)

(0,1,1,2,2,1,2,2,3,3,2,2)=\({\psi_0({\Omega_{\omega}×{\omega}})}\)

(0,1,1,2,2,1,2,2,3,3,2,3,3,4,3)=\({\psi_0({\Omega_{\omega}×{\Omega}})}\)

(0,1,1,2,2,1,2,2,3,3,2,3,3,4,3,2,3,3,4,3,4,4,5,5)=\({\psi_0({\Omega_{\omega}×{\psi_1({\Omega_{\omega}})}})}\)

(0,1,1,2,2,1,2,2,3,3,2,3,3,4,4)=\({\psi_0({\Omega_{\omega}^2})}\)

(0,1,1,2,2,1,2,2,3,3,2,3,3,4,4,2)=\({\psi_0({\Omega_{\omega}^2+1})}\)

(0,1,1,2,2,1,2,2,3,3,2,3,3,4,4,2,1,2,2,3,3,2,3,3,4)=\({\psi_0({\Omega_{\omega}^2}+{\psi_1({\Omega_{\omega}×2})})}\)

(0,1,1,2,2,1,2,2,3,3,2,3,3,4,4,2,2)=\({\psi_0({\Omega_{\omega}^2+{\psi_1({\Omega_{\omega}×2})×{\omega}}})}\)

(0,1,1,2,2,1,2,2,3,3,2,3,3,4,4,2,3,3,4,4,3,4,4,5,5,4,5,5,6,6)=\({\psi_0({\Omega_{\omega}^2+{\psi_1({\Omega_{\omega}^2})}})}\)

(0,1,1,2,2,1,2,2,3,3,2,3,3,4,4,3)=\({\psi_0({\Omega_{\omega}^2+{\psi_1({\Omega_{\omega}^2+1})}})}\)

(0,1,1,2,2,1,2,2,3,3,2,3,3,4,4,3,2,1,2,2,3,3)=\({\psi_0({\Omega_{\omega}^2+{\Omega_{\omega}}})}\)

(0,1,1,2,2,1,2,2,3,3,2,3,3,4,4,3,2,1,2,2,3,3,2,3,3,4,4)=\({\psi_0({\Omega_{\omega}^2×2})}\)

(0,1,1,2,2,1,2,2,3,3,2,3,3,4,4,3,2,2)=\({\psi_0({\Omega_{\omega}^2×{\omega}})}\)

(0,1,1,2,2,1,2,2,3,3,2,3,3,4,4,3,2,3,3,4,4)=\({\psi_0({\Omega_{\omega}^3})}\)

(0,1,1,2,2,1,2,2,3,3,2,3,3,4,4,3,3)=\({\psi_0({\Omega_{\omega}^{\omega}})}\)

(0,1,1,2,2,1,2,2,3,3,2,3,3,4,4,3,4,4,5,5)=\({\psi_0({\Omega_{\omega}^{\Omega_{\omega}}})}\)

(0,1,1,2,2,2)=\({\psi_0({\psi_{\omega}(0)})}\)

ベクレミシェフ行列

表記

• \(k,m,n,a_{x,y}:\)非負整数
• \(S_x=(a_{x,0},a_{x,1}...a_{x,m})\)
• 正規形:\(S_0S_1...S_k[n]\)

計算法

rule1:\(Z[n]=n+1\)

rule2:\(S_0S_1...S_{k-1}Z[n]=S_0S_1...S_{k-1}[n+1]\)

rule3:\(S_0S_1...S_{k-1}(0,0...0,i+1...a_m)[n]=S_0S_1...S_{k-1}(0,0...n,i...a_m)[n]\)

rule4:

• \(r=Max\{b|∃c[((c<m)⇒(∀d,c<d≤m[(a_{b,d}=a_{k,d}]))∧(a_{b,c}<a_{k,c})]\}\)
• \(A=S_0S_1...S_r\)
• \(B=S_{r+1}S_{r+2}...S_{k-1}(a_{k,0}-1,a_{k,1}...a_{k,m})\)
• \(S_0S_1...S_k[n]=A\underbrace{BB...B}_n[n]\)

評価

(0,1)=\({\varepsilon_0}\)

(0,1)(0,0)(0,1)=\({\varepsilon_0×2}\)

(0,1)(1,0)=\({\varepsilon_0×{\omega}}\)

(0,1)(1,0)(0,1)=\({\varepsilon_0^2}\)

(0,1)(2,0)=\({\varepsilon_0^{\omega}}\)

(0,1)(2,0)(0,1)=\({\varepsilon_0^{\varepsilon_0}}\)

(0,1)(3,0)=\({\varepsilon_0^{\varepsilon_0^{\omega}}}\)

(0,1)(0,1)=\({\varepsilon_1}\)

(1,1)=\({\psi_0({\omega})}\)

(1,1)(0,1)=\({\psi_0({\omega+1})}\)

(1,1)(0,1)(1,1)=\({\psi_0({\omega×2})}\)

(1,1)(1,1)=\({\psi_0({\omega^2})}\)

(2,1)=\({\psi_0({\omega^{\omega}})}\)

(0,2)=\({\psi_0({\psi_0(0)})}\)

(0,0,1)=\({\psi_0({\Omega})}\)

(0,0,1)(0,1,0)=\({\psi_0({\Omega+1})}\)

(0,0,1)(0,1,0)(0,0,1)=\({\psi_0({\Omega+{\psi_0({\Omega})}})}\)

(0,0,1)(0,2,0)=\({\psi_0({\Omega+{\psi_0({\Omega+1})}})}\)

(0,0,1)(0,0,1)=\({\psi_0({\Omega×2})}\)

(1,0,1)=\({\psi_0({\Omega×{\omega}})}\)

(0,1,1)=\({\psi_0({\Omega×{\psi_0(0)}})}\)

(0,0,2)=\({\psi_0({\Omega×{\psi_0({\Omega})}})}\)

(0,0,0,1)=\({\psi_0({\Omega^2})}\)

(0,0...0,1)=\({\psi_0({\Omega^{\omega}})}\)

弱行列(仮)

表記

• \(k,m,n,a_{x,y}:\)非負整数
• \(S_x=(a_{x,0},a_{x,1}...a_{x,m})\)
• 正規形:\(S_0S_1...S_k[n]\)

計算法

rule1:\(Z[n]=n+1\)

rule2:\(S_0S_1...S_{k-1}Z[n]=S_0S_1...S_{k-1}[n+1]\)

rule3:

• \(r=Max\{b|a_{b,0}<a_{k,0}\}\)
• \({\Delta}=(d_0,d_1...d_m)\)
• \(d_c=\begin{cases}a_{k,c}-a_{r,c}&\text{if}　a_{r,c+1}<a_{k,c+1}\\0&\text{otherwise}\end{cases}\)
• \(A=S_0S_1...S_{r-1}\)
• \(B_0=S_rS_{r+1}...S_{k-1}\)
• \(B_{b+1}=B_b+{\Delta}\)

\(S_0S_1...S_k[n]=AB_0B_1...B_n[n]\)

評価

(0,0)(1,1)=\({\varepsilon_0}\)

(0,0)(1,1)(0,0)(1,1)=\({\varepsilon_0×2}\)

(0,0)(1,1)(1,0)=\({\varepsilon_0×{\omega}}\)

(0,0)(1,1)(1,0)(2,1)=\({\varepsilon_0^2}\)

(0,0)(1,1)(1,0)(2,1)(2,0)=\({\varepsilon_0^{\omega}}\)

(0,0)(1,1)(1,0)(2,1)(2,0)(3,1)=\({\varepsilon_0^{\varepsilon_0}}\)

(0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)=\({\varepsilon_0^{\varepsilon_0^{\omega}}}\)

(0,0)(1,1)(1,1)=\({\varepsilon_1}\)

(0,0)(1,1)(2,1)=\({\psi_0({\omega})}\)

(0,0)(1,1)(2,1)(1,1)=\({\psi_0({\omega+1})}\)

(0,0)(1,1)(2,1)(1,1)(2,1)=\({\psi_0({\omega×2})}\)

(0,0)(1,1)(2,1)(2,1)=\({\psi_0({\omega^2})}\)

(0,0)(1,1)(2,1)(3,1)=\({\psi_0({\omega^{\omega}})}\)

(0,0)(1,1)(2,2)=\({\psi_0({\psi_0(0)})}\)

(0,0,0)(1,1,1)=\({\psi_0({\Omega})}\)

(0,0,0)(1,1,1)(1,1,0)=\({\psi_0({\Omega+1})}\)

(0,0,0)(1,1,1)(1,1,0)(2,2,1)=\({\psi_0({\Omega+{\psi_0({\Omega})}})}\)

(0,0,0)(1,1,1)(1,1,0)(2,2,1)(2,2,0)=\({\psi_0({\Omega+{\psi_0({\Omega+1})}})}\)

(0,0,0)(1,1,1)(1,1,1)=\({\psi_0({\Omega×2})}\)

(0,0,0)(1,1,1)(2,1,1)=\({\psi_0({\Omega×{\omega}})}\)

(0,0,0)(1,1,1)(2,2,1)=\({\psi_0({\Omega×{\psi_0(0)}})}\)

(0,0,0)(1,1,1)(2,2,2)=\({\psi_0({\Omega×{\psi_0({\Omega})}})}\)

(0,0,0,0)(1,1,1,1)=\({\psi_0({\Omega^2})}\)

(0,0...0)(1,1...1)=\({\psi_0({\Omega^{\omega}})}\)

偽原始数列

小偽原始数列+大SZNO数列

表記

\((a_0,a_1,a_2...a_k)[n]\)

定義

rule1:\((0)[n]=n+1\)

rule2:\((a_0,a_1,a_2...a_{k-1},0)[n]=(a_0,a_1,a_2...a_{k-1})[n+1]\)

rule3:

• \(p(b)=Max\{c|(a_c=0)∧(c<b)\}\)
• \(i=Max\{b|(a_b=0)∧(b-1-p(b-1)<k-p(k))\}\)
• \(A=(a_0,a_1,a_2...a_{i-1})\)
• \(B=(a_i,a_{i+1},a_{i+2}...a_{k-1})\)
• \((a_0,a_1,a_2...a_k)[n]=A\underbrace{\frown B\frown B...B}_n[n]\)

評価

(0,1)=\({\omega}\)

(0,1,0,0,1)=\({\omega×2}\)

(0,1,0,1)=(|0,1,0|0,1,0|0,1,0|...)=\({\omega^2}\)

(0,1,0,1,0,0,1,0,1)=\({\omega^2×2}\)

(0,1,0,1,0,1)=\({\omega^3}\)

(0,1,2)=(|0,1|0,1|0,1|...)=\({\omega^{\omega}}\)

(0,1,2,0,1)=\({\omega^{\omega+1}}\)

(0,1,2,0,1,0,1,2)=\({\omega^{\omega×2}}\)

(0,1,2,0,1,2)=\({\omega^{\omega^2}}\)

(0,1,2,3)=\({\omega^{\omega^{\omega}}}\)

(0,1,2...)=\({\varepsilon_0}\)

小多重階差数列システム

表記

\((a_1,a_2,a_3...a_k)[n]\)

計算法

rule1:\((1)[n]=n+1\)

rule2:\((a_1,a_2,a_3...a_{k-1},1)[n]=(a_1,a_2,a_3...a_{k-1})[n+1]\)

rule3:

• \(a_{0,b}=a_b\)
• \(a_{c+1,b}=a_{c,b+1}-a_{c,b}\)
• \(d=Max\{c|a_{c,k-c}>1\}\)
• \(e=Max\{b|a_{d,b}=1\}\)
• \(A=(a_{d,1},a_{d,2},a_{d,3}...a_{d,e-1})\)
• \(B=(a_{d,e},a_{d,e+1},a_{d,e+2}...a_{d,k-d-1})\)
• \((f_{d,1},f_{d,2},f_{d,3}...)=A\frown \underbrace{B\frown B\frown B...B}_n\)
• \(f_{c,1}=a_{c,1}\)
• \(f_{c,b+1}=f_{c,b}+f_{c+1,b}\)

\((a_1,a_2,a_3...a_k)[n]=(f_{0,1},f_{0,2},f_{0,3}...)[n]\)

評価

(1,2)=(|1|1|1|...)=\({\omega}\)

(1,2,3)=(|1,2|1,2|1,2|...)=\({\omega^2}\)

(1,2,4)=(1[1,2])=(1[|1|1|1|...])=(1,2,3,4...)=\({\omega^{\omega}}\)

(1,2,4,5)=(|1,2,4|1,2,4|1,2,4|...)=\({\omega^{\omega+1}}\)

(1,2,4,5,7)=(1[1,2,1,2])=(1[1,2|1|1|1|...])=(1,2,4,5,6,7...)=\({\omega^{\omega×2}}\)

(1,2,4,7)=(1[1,2,3])=(1[|1,2|1,2|1,2|...])=(1,2,4,5,7,8,10...)=\({\omega^{\omega^2}}\)

(1,2,4,8)=(1[1[1,2]])=(1[1,2,3,4...])=(1,2,4,7,11...)=\({\omega^{\omega^{\omega}}}\)

(1,2,4,8,13)=(1[1,2,4,5])=(1[|1,2,4|1,2,4|1,2,4|...])=(1,2,4,8,9,11,15,16,18,22...)=\({\omega^{\omega^{\omega+1}}}\)

(1,2,4,8,13,20)=(1[1,2,4,5,7])=(1[1,2,4,5,6,7...])=(1,2,4,8,13,19,26...)=\({\omega^{\omega^{\omega×2}}}\)

(1,2,4,8,15)=(1[1,2,4,7])=(1[1,2,4,5,7,8,10])=(1,2,4,8,13,20,28,38...)=\({\omega^{\omega^{\omega^2}}}\)

(1,2,4,8,16)=(1[1,2,4,8])=(1[1,2,4,7,11...])=(1,2,4,8,15,26...)=\({\omega^{\omega^{\omega^{\omega}}}}\)

(1,2,4,8...2^n)=\({\varepsilon_0}\)

小SZ数列(SSZSs)

\(SpS+SZ\)

表記

\((a_0,a_1,a_2...a_k)[n]\

計算法

\(p(x)=\begin{cases}Max\{b|(b<x)∧(a_b<a_x)\}&\text{if}　∃b\\-1&\text{otherwise}\end{cases}\)

• rule1:　\((k=0)∧(a_k=0)\)

\((0)[n]=n^2\)

• rule2:　\((a_k=0)∧(p(k)=-1)\)

\((a_0,a_1,a_2...a_{k-1},a_k)[n]=(a_0,a_1,a_2...a_{k-1})[n^2]\)

• rule3:　\(p(k)=-1\)

\((a_0,a_1,a_2...a_k)[n]=(a_0,a_1,a_2...a_k-1)[n^2]\)

• rule4:　\(a_k-a_{p(k)}=1\)

\(r=p(p(k))+1\)

\(A=(a_0,a_1,a_2...a_{r-1})\)

\(B=(a_r,a_{r+1},a_{r+2}...a_{k-1})\)

\((a_0,a_1,a_2...a_k)[n]=A\frown \{B+n^2\}[n^2]\)

• rule5:

\(r=p(k)\)

\(A=(a_0,a_1,a_2...a_{r-1})\)

\(B=(a_r,a_{r+1},a_{r+2}...a_{k-1},a_k-1)\)

\((a_0,a_1,a_2...a_k)[n]=A\frown \underbrace{B\frown B\frown B...B}_{n^2+1}[n^2]\)

素因数列システム

計算法

• \(p_i:i\)番目の素数(\(p_0=2\))
• \(□:0\)個以上の項
• \(m,j:1\)以上の自然数

rule1:　\(()[n]=n\)

rule2:　\((□,1)[n]=(□)[n+1]\)

rule3:　\((□,m×2)[n]=(□,\underbrace{m,m,...m}_n)[n]\)

rule4:　\((□,m×p_j)[n]=(□,m×p_{j-1}^n)[n]\)

　　　　ただし、\(m\)は\(p_j\)未満の素数で割り切れない。

未定義

小NZ数列システム(SNZSs)

\(L_1+NZ\)

(1)=[1,0]=\({\omega}\)

(1,0)=[1,1]=\({\omega+1}\)

(1,0,0)=[1,2]=\({\omega+2}\)

(0,1)=[2,0]=\({\omega×2}\)

(0,1,0)=[2,1]=\({\omega×2+1}\)

(0,0,1)=[3,0]=\({\omega×3}\)

(1,0,1)=[1,0,0]=\({\omega^2}\)

(1,0,0,1)=[1,1,0]=\({\omega^2+{\omega}}\)

(0,1,0,1)=[2,0,0]=\({\omega^2×2}\)

(1,0,1,0,1)=[1,0,0,0]=\({\omega^3}\)

(1,1)=[1[0]0]=\({\omega^{\omega}}\)

(1,1,0)=[1[0]1]=\({\omega^{\omega}+1}\)

(1,1,0,1)=[1[0],1,0]=\({\omega^{\omega}+{\omega}}\)

(1,1,0,1,0,1)=[1[0]1,0,0]=\({\omega^{\omega}+{\omega^2}}\)

(0,1,1)=[2[0]0]=\({\omega^{\omega}×2}\)

(1,0,1,1)=[1,0[0]0]=\({\omega^{\omega+1}}\)

(1,1,0,1,1)=[1[0]0[0]0]=\({\omega^{\omega×2}}\)

(1,1,1)=[1[1]0]=\({\omega^{\omega^2}}\)

(2)=[1[1,0]0]=\({\omega^{\omega^{\omega}}}\)

(2,0,1)=[1[1,0]1,0]=\({\omega^{\omega^{\omega}}+{\omega}}\)

(2,0,1,1)=[1[1,0]1[0]0]=\({\omega^{\omega^{\omega}}+{\omega^{\omega}}}\)

(0,2)=[2[1,0]0]=\({\omega^{\omega^{\omega}}×2}\)

(2,0,2)=[1[1,0]0[1,0]0]=\({\omega^{\omega^{\omega}×2}}\)

(2,1)=[1[1,1]0]=\({\omega^{\omega^{\omega+1}}}\)

(1,2)=[1[2,0]0]=\({\omega^{\omega^{\omega×2}}}\)

(2,1,2)=[1[1,0,0]0]=\({\omega^{\omega^{\omega^2}}}\)

(2,2)=[1[1[0]0]0]=\({\omega^{\omega^{\omega^{\omega}}}}\)

(3)=\({\omega^{\omega^{\omega^{\omega^{\omega}}}}}\)

(n)=\({\varepsilon_0}\)

第2小SZ数列(SSSZSs)

\(SPpS+SZ\)

((0,1)=\({\omega}\)

(0,1,0,2)=\({\omega^2}\)

(0,1,0,2,0,3,...)=(0,1,1)=\({\omega^{\omega}}\)

(0,1,1,0,2)=\({\omega^{\omega+1}}\)

(0,1,1,0,2,2)=\({\omega^{\omega^2}}\)

(0,1,1,1)=\({\omega^{\omega^{\omega}}}\)

(0,1,2)=\({\varepsilon_0}\)

(0,1,2,0,2)=\({\varepsilon_0}×{\omega}\)

(0,1,2,0,2,0,2,3)=\({\varepsilon_0^2}\)

(0,1,2,0,2,2)=\({\varepsilon_0^{\omega}}\)

(0,1,2,0,2,2,0,2,3)=\({\varepsilon_0^{\varepsilon_0}}\)

(0,1,2,0,2,3)=\({\varepsilon_1}\)

(0,1,2,0,2,3,0,3,4)=\({\varepsilon_2}\)

(0,1,2,1)=\({\psi_0({\omega})}\)

(0,1,2,1,0,2,3)=\({\psi_0({\omega+1})}\)

(0,1,2,1,0,2,3,2)=\({\psi_0({\omega}^2)}\)

(0,1,2,1,1)=\({\psi_0({\omega^{\omega}})}\)

(0,1,2,1,2)=\({\psi_0({\psi_0(0)})}\)

(0,1,2,1,3)=\({\psi_0({\Omega})}\)

(0,1,2,2)=\({\psi_0({\Omega^{\omega}})}\)

(0,1,2,3)=\({\psi_0({\Omega^{\varepsilon_0}})}\)

(0,1,2,3,...n)=\({\psi_0({\Omega^{\Omega}})}\)

大SZ数列(LSZSs)

\(pS+SZ\)、\({\Omega}\)行BM?

(0,1,1)=(|0,1|0,2|0,3|...)=\({\varepsilon_0}\)

(0,1,1,0,2,1)=(0,1,1|0,2|0,3|0,4|...)=\({\varepsilon_0^2}\)

(0,1,1,0,2,1,0,3,1)=(0,1,1,0,2,1|0,3|0,4|0,5|...)=\({\varepsilon_0^{\varepsilon_0}}\)

(0,1,1,0,2,2)=(|0,1,1|0,2,1|0,3,1|...)=\({\varepsilon_1}\)

(0,1,1,0,2,2,0,2,2)=(|0,1,1,0,2,2|0,2,1,0,3,2|0,3,1,0,3,2|...)=\({\varepsilon_2}\)

(0,1,1,0,2,2,0,3,2,)=(|0,1,1,0,2,2|0,3,1,0,4,2|0,5,1,0,6,2|...)=\({\psi_0({\Omega})}\)

(0,1,1,0,2,2,0,3,3)=(0,1,1|0,2,2|0,3,2|0,4,2|...)=\({\psi_0({\psi_1(0)})}\)

(0,1,1,1)=(|0,1,1|0,2,2|0,3,3|...)=\({\psi_0({\Omega_{\omega}})}\)=B(0,0,0)(1,1,1)

(0,1,1,1,0,2)=(|0,1,1,1|0,1,1,1|0,1,1,1|...)=B(0,0,0)(1,1,1)(1,0,0)

(0,1,1,1,0,2,2)=(|0,1,1,1|0,2,1,1|0,3,1,1|...)=B(0,0,0)(1,1,1)(1,1,0)

(0,1,1,1,0,2,2,1)=(0,1,1,1|0,2,2|0,3,3|0,4,4|...)=B(0,0,0)(1,1,1)(1,1,0)(2,2,1)

(0,1,1,1,0,2,2,2)=(0,1,1,1|0,2,2,1|0,3,3,1|...)=B(0,0,0)(1,1,1)(1,1,1)

(0,1,1,1,0,2,2,2,0,3)=(0,1,1,1|0,2,2,2|0,2,2,2|0,2,2,2|...)=B(0,0,0)(1,1,1)(2,0,0)

(0,1,1,1,0,2,2,2,0,3,3,3)=(0,1,1,1|0,2,2,2|0,3,3,2|0,4,4,2|...)=B(0,0,0)(1,1,1)(2,2,2)

(0,1,1,1,1)=(|0,1,1,1|0,2,2,2|0,3,3,3|...)=B(0,0,0,0)(1,1,1,1)

(0,1,2)=(0|1|1|1|...)=(0,0,0,...)(1,1,1,...)

ペア大SZNO数列システム(PLSZNOSs)

(0,0)(1,1)=\({\beta_0}({\beta_0}({\Omega}))\)

(0,0)(1,1)(0,0)(0,0)(1,1)=\({\beta_0}({\beta_0}({\Omega}),0,{\beta_0}({\Omega}))\)

(0,0)(1,1)(0,0)(1,0)=\({\beta_0}({\beta_0}({\Omega}),1)\)

(0,0)(1,1)(0,0)(1,0)(1,0)=\({\beta_0}({\beta_0}({\Omega}),1,2)\)

(0,0)(1,1)(0,0)(1,0)(2,1)=\({\beta_0}({\beta_0}({\Omega}),1,{\beta_0}({\Omega}))\)

(0,0)(1,1)(0,0)(1,0)(2,1)(0,0)(1,0)=\({\beta_0}({\beta_0}({\Omega}),1,{\beta_0}({\Omega}),1)\)

(0,0)(1,1)(0,0)(1,0)(2,1)(0,0)(1,0)(1,0)=\({\beta_0}({\beta_0}({\Omega}),2)\)

(0,0)(1,1)(0,0)(1,0)(2,1)(0,0)(1,0)(2,1)=\({\beta_0}({\beta_0}({\Omega}),{\beta_0}({\Omega}))\)

(0,0)(1,1)(0,0)(1,0)(2,1)(1,0)=\({\beta_0}({\beta_0}({\Omega},0))\)

(0,0)(1,1)(0,0)(1,0)(2,1)(1,0)(2,0)(3,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\beta_0}({\Omega}))))\)

(0,0)(1,1)(0,0)(1,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega})))\)

(0,0)(1,1)(0,0)(1,1)(0,0)(1,0)(2,1)(1,0)(2,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega})),1,{\beta_0}({\Omega},0,{\beta_0}({\Omega})))\)

(0,0)(1,1)(0,0)(1,1)(0,0)(1,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega}),0,{\beta_0}({\Omega})))\)

(0,0)(1,1)(1,0)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega}),1))\)

(0,0)(1,1)(1,0)(0,0)(1,0)(0,0)(1,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega}),1,0,{\beta_0}({\Omega})))\)

(0,0)(1,1)(1,0)(0,0)(1,0)(0,0)(1,1)(1,0)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega}),1,0,{\beta_0}({\Omega}),1))\)

(0,0)(1,1)(1,0)(0,0)(1,0)(1,0)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega}),1,1))\)

(0,0)(1,1)(1,0)(0,0)(1,0)(1,0)(1,0)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega}),1,2))\)

(0,0)(1,1)(1,0)(0,0)(1,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega}),1,{\beta_0}({\Omega})))\)

(0,0)(1,1)(1,0)(0,0)(1,1)(0,0)(1,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega}),1,{\beta_0}({\Omega}),0,{\beta_0}({\Omega})))\)

(0,0)(1,1)(1,0)(0,0)(1,1)(1,0)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega}),1,{\beta_0}({\Omega}),1))\)

(0,0)(1,1)(1,0)(1,0)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega}),2))\)

(0,0)(1,1)(1,0)(2,0)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega}),{\omega}))\)

(0,0)(1,1)(1,0)(2,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega}),{\beta_0}({\Omega})))\)

(0,0)(1,1)(1,0)(2,1)(1,0)(2,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega}),{\beta_0}({\Omega}),{\beta_0}({\Omega})))\)

(0,0)(1,1)(1,0)(2,1)(2,0)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0)))\)

(0,0)(1,1)(1,0)(2,1)(2,0)(3,0)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,1)))\)

(0,0)(1,1)(1,0)(2,1)(2,0)(3,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega},0,{\beta_0}({\Omega}))))\)

(0,0)(1,1)(1,1)=\({\beta_0}({\beta_0}({\Omega},0,{\beta_0}({\Omega})))\)

(0,0)(1,1)(2,0)=\({\beta_0}({\beta_0}({\Omega},1))\)

(0,0)(1,1)(2,0)(1,1)=\({\beta_0}({\beta_0}({\Omega},1,0,{\Omega}))\)

(0,0)(1,1)(2,0)(1,1)(1,1)(2,0)=\({\beta_0}({\beta_0}({\Omega},1,0,{\Omega},1))\)

(0,0)(1,1)(2,0)(1,1)(2,0)=\({\beta_0}({\beta_0}({\Omega},1,1))\)

(0,0)(1,1)(2,0)(2,0)=\({\beta_0}({\beta_0}({\Omega},1,2))\)

(0,0)(1,1)(2,0)(3,1)=\({\beta_0}({\beta_0}({\Omega},1,{\beta_0}({\Omega})))\)

(0,0)(1,1)(2,1)=\({\beta_0}({\beta_0}({\Omega},1,{\Omega}))\)

(0,0)(1,1)(2,1)(1,1)(2,0)=\({\beta_0}({\beta_0}({\Omega},1,{\Omega},1))\)

(0,0)(1,1)(2,1)(1,1)(2,1)=\({\beta_0}({\beta_0}({\Omega},1,{\Omega},1,{\Omega}))\)

(0,0)(1,1)(2,1)(2,0)=\({\beta_0}({\beta_0}({\Omega},2))\)

(0,0)(1,1)(2,1)(2,0)(1,1)(1,1)(2,1)(2,0)=\({\beta_0}({\beta_0}({\Omega},2,0,{\Omega},2))\)

(0,0)(1,1)(2,1)(2,0)(1,1)(2,0)=\({\beta_0}({\beta_0}({\Omega},2,1))\)

(0,0)(1,1)(2,1)(2,0)(1,1)(2,0)(1,1)(2,1)=\({\beta_0}({\beta_0}({\Omega},2,1,{\Omega}))\)

(0,0)(1,1)(2,1)(2,0)(1,1)(2,0)(1,1)(2,1)(2,0)=\({\beta_0}({\beta_0}({\Omega},2,1,{\Omega},2))\)

(0,0)(1,1)(2,1)(2,0)(1,1)(2,0)(2,0)=\({\beta_0}({\beta_0}({\Omega},2,2))\)

(0,0)(1,1)(2,1)(2,0)(1,1)(2,1)=\({\beta_0}({\beta_0}({\Omega},2,{\Omega}))\)

(0,0)(1,1)(2,1)(2,0)(1,1)(2,1)(2,0)=\({\beta_0}({\beta_0}({\Omega},2,{\Omega},2))\)

(0,0)(1,1)(2,1)(2,0)(2,0)=\({\beta_0}({\beta_0}({\Omega},3))\)

(0,0)(1,1)(2,1)(2,0)(3,0)=\({\beta_0}({\beta_0}({\Omega},{\omega}))\)

(0,0)(1,1)(2,1)(2,1)=\({\beta_0}({\beta_0}({\Omega},{\Omega}))\)

(0,0)(1,1)(2,1)(3,0)=\({\beta_0}({\beta_0}({\beta_1(0)}))\)

偽バシク行列

(0,0)(1,1)=\({\varepsilon_0}\)

(0,0)(1,1)(0,0)(1,1)=|(0,0)(1,1)(0,0)|(1,0)(2,1)(1,0)|(2,0)(3,1)(2,0)|...=\({\varepsilon_1}\)

(0,0)(1,1)(2,0)=|(0,0)(1,1)|(0,0)(1,1)|(0,0)(1,1)|...=\({\psi_0({\omega})}\)

(0,0)(1,1)(2,0)(3,0)(4,1)=\({\psi_0({\psi_0(0)})}\)

(0,0)(1,1)(2,0)(3,1)=\({\psi_0({\Omega})}\)

(0,0)(1,1)(2,0)(3,1)(0,0)(1,1)=\({\psi_0({\Omega+1})}\)

(0,0)(1,1)(2,0)(3,1)(0,0)(1,1)(0,0)(1,1)(2,0)(3,1)=\({\psi_0({\Omega×2})}\)

(0,0)(1,1)(2,0)(3,1)(0,0)(1,1)(2,0)=\({\psi_0({\Omega×{\omega}})}\)

(0,0)(1,1)(2,0)(3,1)(0,0)(1,1)(2,0)(3,1)=\({\psi_0({\Omega^2})}\)

(0,0)(1,1)(2,0)(3,1)(4,0)=\({\psi_0({\Omega^{\omega}})}\)

(0,0)(1,1)(2,0)(3,1)(4,0)(5,1)=\({\psi_0({\Omega^{\Omega}})}\)

(0,0)(1,1)(2,2)=|(0,0)(1,0)|(2,0)(3,1)|(4,0)(5,1)|...=\({\psi_0({\psi_1(0)})}\)

(0,0)(1,1)(2,2)(0,0)(1,1)(2,0)(3,1)(4,2)=\({\psi_0({\psi_1(0)×2})}\)

(0,0)(1,1)(2,2)(0,0)(1,1)(2,2)=\({\psi_0({\psi_1(1)})}\)

(0,0)(1,1)(2,2)(3,0)(4,1)=\({\psi_0({\psi_1({\Omega})})}\)

(0,0)(1,1)(2,2)(3,0)(4,1)(5,0)(6,1)(7,2)=\({\psi_0({\psi_1({\psi_1(0)})})}\)

(0,0)(1,1)(2,2)(3,0)(4,1)(5,2)=\({\psi_0({\psi_1({\Omega_2})})}\)

(0,0)(1,1)(2,2)(3,3)=\({\psi_0({\psi_1({\psi_2(0)})})}\)

(0,0,0)(1,1,1)=\({\psi_0({\Omega_{\omega}})}\)

(0,0,0)(1,1,1)(0,0,0)(1,1,1)

=|(0,0,0)(1,1,1)(0,0,0)|(1,1,0)(2,2,1)(1,1,0)|(2,2,0)(3,3,1)(2,2,0)|...

(0,0,0)(1,1,1)(0,0,0)(1,1,0)=\({\psi_0({\Omega_{\omega}+1})}\)

(0,0,0)(1,1,1)(0,0,0)(1,1,0)(2,2,1)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}})}})}\)

(0,0,0)(1,1,1)(0,0,0)(1,1,1)=\({\psi_0({\Omega_{\omega}×2})}\)

(0,0,0)(1,1,1)(2,0,0)=\({\psi_0({\Omega_{\omega}×{\omega}})}\)

(0,0,0)(1,1,1)(2,0,0)(0,0,0)(1,1,1)

=|(0,0,0)(1,1,1)(2,0,0)(0,0,0)|(1,1,0)(2,2,1)(3,0,0)(1,1,0)|(2,2,0)(3,3,1)(4,0,0)(2,2,0)|...

2次行列

(0)(1)(3)(5)=(0,0)|(1,0)(3,0)|(4,3)(6,0)|(7,6)(9,0)|...

(0,0)(1,0)(3,0)(4,3)=(0,0,0)|(1,0,0)(3,0,0)|(4,2,2)(6,0,0)|(7,4,4)(9,0,0)|...

(0,0,0)(1,0,0)(3,0,0)(4,2,2)=(0,0,0,0)|(1,0,0,0)(3,0,0,0)|(4,2,1,1)(6,0,0)|(7,4,2,2)(9,0,0,0)|...

(0)(1)(3)(6)=(0,0)(1,0)|(3,0)|(5,2)|(7,4)|...

(0,0)(1,0)(3,0)(5,1)=(0)(1)|(3)|(5)|(7)|...

(0)(1)(3)=(0,0)|(1,0)|(2,1)|(3,2)|...=B(0,0,0)(1,1,1)

(0)(1)(3)(3)=(0,0)|(1,0)(3,0)|(2,1)(4,0)|(3,2)(5,0)|...=B(0,0,0)(1,1,1)(1,1,1)

(0)(1)(3)(4)=B(0,0,0)(1,1,1)(2,0,0)

(0,0)(1,0)(3,0)(4,1)=(0,0)|(1,0)(3,0)|(4,0)(6,0)|(7,0)(9,0)|...=B(0,0,0)(1,1,1)(2,1,0)

(0,0)(1,0)(3,0)(4,1)(3,0)=(0,0)|(1,0)(3,0)(4,1)|(2,1)(4,0)(5,2)|(3,2)(5,0)(6,3)|...

(0,0)(1,0)(3,0)(4,1)(3,0)(4,1)=B(0,0,0)(1,1,1)(2,1,0)(1,1,1)(2,1,0)

(0,0)(1,0)(3,0)(4,1)(4,0)=B(0,0,0)(1,1,1)(2,1,0)(2,0,0)

(0,0)(1,0)(3,0)(4,1)(5,0)=B(0,0,0)(1,1,1)(2,1,0)(3,0,0)

(0,0)(1,0)(3,0)(4,1)(6,0)=(0,0)(1,0)(3,0)|(4,1)|(5,2)|(6,3)|...=B(0,0,0)(1,1,1)(2,1,0)(3,2,1)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)=(0,0)|(1,0)(3,0)|(4,1)(6,0)|(7,2)(9,0)|...=B(0,0,0)(1,1,1)(2,1,1)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(3,0,0)=(0,0,0)|(1,0,0)(3,0,0)(4,1,1)|(2,1,0)(4,0,0)(5,2,1)|(3,2,0)(5,0,0)(6,3,1)|...=B(0,0,0)(1,1,1)(2,1,1)(1,1,1)

(0,0)(1,0)(3,0)(4,2)=(0,0,0)|(1,0,0)(3,0,0)|(4,1,1)(6,0,0)|(7,2,2)(9,0,0)|...

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(6,0,0)=(0,0,0)(1,0,0)(3,0,0)|(4,1,1)|(5,2,1)|(6,3,1)|...

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(5,0,0)=B(0,0,0)(1,1,1)(2,1,1)(3,0,0)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(5,1,0)=B(0,0,0)(1,1,1)(2,1,1)(3,1,0)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(5,1,1)=B(0,0,0)(1,1,1)(2,1,1)(3,1,1)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(5,2,0)=B(0,0,0)(1,1,1)(2,2,0)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(5,2,0)(7,0,0)=B(0,0,0)(1,1,1)(2,2,0)(3,3,1)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(5,2,1)=B(0,0,0)(1,1,1)(2,2,1)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(5,2,1)(3,0,0)=B(0,0,0)(1,1,1)(2,2,1)(1,1,1)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(5,2,1)(3,0,0)(4,1,1)(5,2,1)=B(0,0,0)(1,1,1)(2,2,1)(1,1,1)(2,2,1)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(5,2,1)(4,1,1)=B(0,0,0)(1,1,1)(2,2,1)(2,1,1)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(5,2,1)(4,1,1)(5,2,1)=B(0,0,0)(1,1,1)(2,2,1)(2,1,1)(3,2,1)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(5,2,1)(5,2,1)=B(0,0,0)(1,1,1)(2,2,1)(2,2,1)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(5,2,1)(6,0,0)=B(0,0,0)(1,1,1)(2,2,1)(3,0,0)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(5,2,1)(6,3,1)=B(0,0,0)(1,1,1)(2,2,1)(3,3,1)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(6,0,0)=B(0,0,0)(1,1,1)(2,2,2)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(6,0,0)(4,1,1)(6,0,0)=B(0,0,0)(1,1,1)(2,2,2)(2,1,1)(3,2,2)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(6,0,0)(6,0,0)=B(0,0,0)(1,1,1)(2,2,2)(2,2,2)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(6,0,0)(7,0,0)=B(0,0,0)(1,1,1)(2,2,2)(3,0,0)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(6,0,0)(7,1,0)=B(0,0,0)(1,1,1)(2,2,2)(3,1,0)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(6,0,0)(7,1,1)(8,2,0)=B(0,0,0)(1,1,1)(2,2,2)(3,1,1)(4,2,0)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(6,0,0)(7,1,1)(9,0,0)=B(0,0,0)(1,1,1)(2,2,2)(3,1,1)(4,2,2)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(6,0,0)(7,2,0)=B(0,0,0)(1,1,1)(2,2,2)(3,2,0)

(0,0,0)(1,0,0)(3,0,0)(4,1,1)(6,0,0)(7,2,2)(9,0,0)=B(0,0,0)(1,1,1)(2,2,2)(3,3,3)

(0,0)(1,0)(3,0)(4,2)=B(0,0,0,0)(1,1,1,1)

(0,0)(1,0)(3,0)(5,1)=B(0(0)0)(1(0)1)

(0,0)(1,0)(3,0)(5,1)(3,0)=(0,0)|(1,0)(3,0)(5,1)|(2,1)(4,0)(6,2)|(3,2)(5,0)(7,3)|...

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,2)=(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)|(4,0,0)|(6,1,1)|(8,2,2)|...

(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)(4,0,0)(6,1,1)=(0,0)(1,0)(3,0)(5,1)(2,1)|(4,0)|(6,1)|(8,2)|...

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(8,2)=(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)|(6,1)|(8,1)|(10,1)|...

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(8,1)=(0,0)(1,0)(3,0)(5,1)(2,1)|(4,0)(6,1)|(8,0)(10,1)|(12,0)(14,1)|...

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(8,0)=(0,0)(1,0)(3,0)(5,1)|(2,1)(4,0)(6,1)|(7,6)(9,0)(11,1)|(12,11)(14,0)(16,1)|...

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)=B(0,0,0(0)0)(1,1,1(0)1)(1,1,0(0)0)(2,2,1(0)1)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(4,0)=(0,0)(1,0)(3,0)(5,1)|(2,1)(4,0)(6,1)|(3,2)(5,0)(7,1)|(4,3)(6,0)(8,1)|...=B(0,0,0(0)0)(1,1,1(0)1)(1,1,1(0)0)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(4,0)(5,1)=B(0,0,0(0)0)(1,1,1(0)1)(1,1,1(0)0)(2,1,0(0)0)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(4,0)(5,1)(7,0)(9,1)=B(0,0,0)(1,1,1(0)1)(1,1,1(0)0)(2,1,0(0)0)(3,2,1(0)1)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(4,0)(6,0)=(0,0)(1,0)(3,0)(5,1)|(2,1)(4,0)(6,1)(4,0)|(5,4)(7,0)(9,1)(7,0)|(8,7)(10,0)(12,1)(10,0)|...

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(4,0)(5,2)=B(0,0,0(0)0)(1,1,1(0)1)(1,1,1(0)0)(2,1,0(0)0)(3,2,1(0)1)(3,2,1(0)0)(4,2,0(0)0)

(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)(4,0,0)(6,1,0)(4,0,0)(5,2,1)=B(0,0,0(0)0)(1,1,1(0)1)(1,1,1(0)0)(2,1,1(0)0)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(4,0)(5,3)=B(0,0,0,0(0)0)(1,1,1,1(0)1)(1,1,1,0(0)0)(2,2,2,2(0)0)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(4,0)(6,1)=B(0(0)0)(1(0)1)(1(0)1)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(5,0)=B(0(0)0)(1(0)1)(2(0)0)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(5,1)=B(0,0(0)0)(1,1(0)1)(2,1(0)0)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(5,1)(7,0)=B(0,0,0(0)0)(1,1,1(0)1)(2,1,0(0)0)(3,2,1(0)0)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(5,1)(7,0)(9,1)=B(0,0,0(0)0)(1,1,1(0)1)(2,1,0(0)0)(3,2,1(0)1)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(5,2)=B(0,0,0(0)0)(1,1,1(0)1)(2,1,0(0)0)(3,2,1(0)1)(4,2,0(0)0)

(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)(4,0,0)(6,1,0)(5,2,1)=B(0,0,0(0)0)(1,1,1(0)1)(2,1,1(0)0)

(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)(4,0,0)(6,1,0)(5,2,1)(7,0,0)(9,1,0)=B(0,0,0,0(0)0)(1,1,1,1(0)1)(2,1,1,0(0)0)(3,2,2,1(0)1)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(5,3)=B(0,0,0,0(0)0)(1,1,1,1(0)1)(2,1,1,1(0)0)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(6,0)(8,1)=B(0,0(0)0)(1,1(0)1)(2,1(0)1)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(6,1)=B(0,0(0)0)(1,1(0)1)(2,2(0)0)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(7,1)=B(0,0(0)0)(1,1(0)1)(2,2(0)0)(3,1(0)0)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(8,0)(10,1)=B(0,0,0(0)0)(1,1,1(0)1)(2,2,1(0)1)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(8,1)=B(0(0)0)(1(0)1)(2(0)1)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(8,1)(4,0)(6,1)(8,1)=B(0(0)0)(1(0)1)(2(0)1)(1(0)1)(2(0)1)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(8,1)(6,0)(8,1)=B(0,0(0)0)(1,1(0)1)(2,2(0)1)(2,1(0)1)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(8,1)(6,1)=B(0,0(0)0)(1,1(0)1)(2,2(0)1)(2,2(0)0)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(8,1)(6,1)(8,0)(10,1)=B(0,0,0(0)0)(1,1,1(0)1)(2,2,2(0)1)(2,2,1(0)1)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(8,1)(6,1)(8,1)=B(0(0)0)(1(0)1)(2(0)1)(2(0)1)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(8,1)(8,0)(10,1)=B(0,0(0)0)(1,1(0)1)(2,2(0)1)(3,1(0)1)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(8,1)(8,1)=B(0,0(0)0)(1,1(0)1)(2,2(0)1)(3,2(0)0)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(8,1)(10,0)(12,1)(14,1)=B(0,0(0)0)(1,1(0)1)(2,2(0)1)(3,2(0)1)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(8,1)(10,1)=B(0(0)0)(1(0)1)(2(0)1)(3(0)1)

(0,0)(1,0)(3,0)(5,1)(2,1)(4,0)(6,1)(8,2)=B(0(0)0)(1(0)1)(2(0)2)

(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)(4,0,0)(6,1,1)=B(0(0)0,0)(1(0)1,1)

(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)(4,0,0)(6,1,1)(4,0,0)(6,1,0)=B(0(0)0,0)(1(0)1,1)(1(0)1,0)

(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)(4,0,0)(6,1,1)(4,0,0)(6,1,1)=B(0(0)0,0)(1(0)1,1)(1(0)1,0)(2(0)2,1)

(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)(4,0,0)(6,1,1)(5,0,0)=B(0(0)0,0)(1(0)1,1)(1(0)1,0)(2(0)2,1)(2(0)0,0)

(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)(4,0,0)(6,1,1)(5,1,0)=B(0,0(0)0,0)(1,1(0)1,1)(1,1(0)1,0)(2,2(0)2,1)(2,1(0)0,0)

(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)(4,0,0)(6,1,1)(6,1,0)=B(0,0(0)0,0)(1,1(0)1,1)(1,1(0)1,0)(2,2(0)2,1)(2,2(0)0,0)

(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)(4,0,0)(6,1,1)(6,1,0)(8,2,1)=B(0(0)0,0)(1(0)1,1)(1(0)1,0)(2(0)2,1)(2(0)2,0)(3(0)3,1)

(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)(4,0,0)(6,1,1)(6,1,1)=B(0(0)0,0)(1(0)1,1)(1(0)1,1)

(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)(4,0,0)(6,1,1)(7,1,0)=B(0,0(0)0,0)(1,1(0)1,1)(2,1(0)0,0)

(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)(4,0,0)(6,1,1)(8,0,0)(10,1,0)=B(0,0(0)0,0)(1,1(0)1,1)(2,1(0)1,0)

(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)(4,0,0)(6,1,1)(8,0,0)(10,1,0)(9,1,0)=B(0,0(0)0,0)(1,1(0)1,1)(2,1(0)1,0)(3,1(0)0,0)

(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)(4,0,0)(6,1,1)(8,0,0)(10,1,0)(10,0,0)(12,1,0)=B(0,0(0)0,0)(1,1(0)1,1)(2,1(0)1,0)(3,1(0)1,0)

(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)(4,0,0)(6,1,1)(8,0,0)(10,1,0)(10,1,0)=B(0,0(0)0,0)(1,1(0)1,1)(2,1(0)1,0)(3,2(0)0,0)

(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)(4,0,0)(6,1,1)(8,0,0)(10,1,0)(12,0,0)(14,1,0)=B(0,0,0(0)0,0)(1,1,1(0)1,1)(2,1,1(0)1,0)(3,2,1(0)1)

(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)(4,0,0)(6,1,1)(8,0,0)(10,1,0)(12,1,0)=B(0,0(0)0,0)(1,1(0)1,1)(2,1(0)1,0)(3,2(0)1,0)

(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)(4,0,0)(6,1,1)(8,0,0)(10,1,0)(12,2,0)=B(0,0(0)0,0)(1,1(0)1,1)(2,1(0)1,0)(3,2(0)2,0)

(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)(4,0,0)(6,1,1)(8,0,0)(10,1,1)=B(0,0(0)0,0)(1,1(0)1,1)(2,1(0)1,0)(3,2(0)1,1)

(0,0,0)(1,0,0)(3,0,0)(5,1,0)(2,1,0)(4,0,0)(6,1,1)(8,1,0)=B(0,0(0)0,0)(1,1(0)1,1)(2,2(0)0,0)

小Y数列

小多重階差数列の拡張

(1,3)=(1[2])=(1[1[1[...]]])=(1,2,4...)=\({\varepsilon_0}\)

(1,3,4)=(|1,3|1,3|1,3|...)=\({\varepsilon_0×{\omega}}\)

(1,3,4,7)=(1[2,1[2]])=(1[2,1[1[1[...]]]])=(1,3,4,6,10...)=\({\varepsilon_0^2}\)

(1,3,4,7,11,18)=(1[2,1[2,1[2]]])=\({\varepsilon_0^{\varepsilon_0}}\)

(1,3,5)=(1[2,2])=(1[2,1[2,1[2...]]])=(1,3,4,7,11,18...)=\({\varepsilon_1}\)

(1,3,6)=(1[2[1]])=(1[2,2,2...])=(1,3,5,7...)=\({\psi_0({\omega})}\)

(1,3,6,12)=(1[2[1[2]]])=\({\psi_0({\psi_0(0)})}\)

(1,3,8)=(1[2[3]])=(1[2[1[2[1[2[...]]]]]])=\({\psi_0({\Omega})}\)

(1[2[3],2])=\({\psi_0({\Omega+1})}\)

(1[2[3],2[3]])=\({\psi_0({\Omega×2})}\)

(1[2[3,1]])=\({\psi_0({\Omega×{\omega}})}\)

(1[2[3,3]])=\({\psi_0({\Omega^2})}\)

(1[2[3[1]]])=\({\psi_0({\Omega^{\omega}})}\)

(1[2[3[4]]])=\({\psi_0({\Omega^{\Omega}})}\)

(1,3,9)=(1[2[4]])=(1[1,2]₂)=\({\psi_0({\psi_1(0)})}\)

(1[2[4],2])=(1[2[4],1[2[4],1[2[4]...]]])=\({\psi_0({\psi_1(0)+1})}\)

(1[2[4],2[3]])=(1[2[4],2[(1[2[4],2[(1[2[4],2[...]])]])]])=\({\psi_0({\psi_1(0)+{\Omega}})}\)

(1[2[4],2[4]])=(1[2[4],2[3[4[...]]]])=\({\psi_0({\psi_1(0)×2})}\)

(1[2[4,1]])=\({\psi_0({\psi_1(0)×{\omega}})}\)

(1[1,2,1]₂)=(1[2[4[5]]])=(1[2[4[1[2[4[1[2[4[...]]]]]]]]])

(1[2[4,3]])=(1[2[4,1[2[4,1[2[4...]]]]]])=\({\psi_0({\psi_1(0)×{\Omega}})}\)

(1[2[4,3[5]]])=\({\psi_0({\psi_1(0)^2})}\)

(1[2[4,4]])=\({\psi_0({\psi_1(1)})}\)

(1[2[4,5]])=\({\psi_0({\psi_1({\omega})})}\)

(1[2[4[5]]])=\({\psi_0({\psi_1({\Omega})})}\)

(1[1,2,1,2]₂)=\({\psi_0({\psi_1({\psi_1(0)})})}\)

(1,3,9,26)=(1[2[4[7]]])=(1[1,2,3]₂)=(1[1,2,1,2,1,2...]₂)=(1[2[4[5[7[8[10[...]]]]]]])=\({\psi_0({\psi_1({\Omega_2})})}\)

(1[2[4[7],4[7]]])=\({\psi_0({\psi_1({\Omega_2×2})})}\)

(1[2[4[7,4]]])=\({\psi_0({\psi_1({\Omega_2×{\Omega}})})}\)

(1[2[4[7,7]]])=\({\psi_0({\psi_1({\Omega_2^2})})}\)

(1[2[4[7[1]]]])=\({\psi_0({\psi_1({\Omega_2^{\omega}})})}\)

(1[2[4[7[8]]]])=(1[1,2,3,1]₂)=\({\psi_0({\psi_1({\Omega_2^{\Omega}})})}\)

(1[1,2,3,1,2]₂)=\({\psi_0({\psi_1({\Omega_2^{\psi_1(0)}})})}\)

(1[1,2,3,4]₂)=(1[2[4[7[11]]]])=\({\psi_0({\psi_1({\Omega_2^{\Omega_2}})})}\)

(1[1[1,2]₁]₂)=\({\psi_0({\psi_1({\psi_2(0)})})}\)

小変形Y数列

(1,2,5)=(1[1[2]])=(1[1[1]₂]₁)=(1[1[1[1[...]]]])=(1,2,4,8...)=\({\varepsilon_0}\)

(1,2,5,10)=(1[1[2,2]])=(1[1[2,1[2,1[2...]]]])=\({\varepsilon_1}\)

(1,2,5,11)=(1[1[2[1]]])=\({\psi_0({\omega})}\)

(1,2,5,13)=(1[1[2[3]]])=\({\psi_0({\Omega})}\)

(1,2,5,14)=(1[1[2[4]]])=(1[1[1,2]₂]₁)=\({\psi_0({\psi_1(0)})}\)

(1,2,5,14,33)=(1[1[2[4,4]]])=\({\psi_0({\psi_1(1)})}\)

(1,2,5,14,40)=(1[1[2[4[7]]]])=(1[1[1,2,3]₂]₁)=\({\psi_0({\psi_1({\Omega_2})})}\)

(1,2,5,14,40,112)=(1[1[2[4[7[11]]]]])=(1[1[1,2,3,4]₂]₁)=\({\psi_0({\psi_1({\Omega_2^{\Omega_2}})})}\)

(1,2,5,14,41)=(1[1[2[4[8]]]])=(1[1[1,2,4]₂]₁)=(1[1[1[1,2]₁]₂]₁)=\({\psi_0({\psi_1({\psi_2(0)})})}\)

変形Y数列

[2]

(1,2,4)=(1[1,2])=(1,2,3,4...)=\({\varepsilon_0}\)

(1,2,4,2)=(|1,2,4|1,2,4|1,2,4|...)=\({\varepsilon_0×{\omega}}\)

(1,2,4,2,4)=(1,2,4|2,3,4|,2,3,4|...)=\({\varepsilon_0^2}\)

(1,2,4,3)=\({\varepsilon_0^{\omega}}\)

(1,2,4,3,5)=\({\varepsilon_0^{\varepsilon_0}}\)

(1,2,4,4)=(1|2,4|3,5|4,6|...)=\({\varepsilon_1}\)

(1,2,4,5)=\({\varepsilon_{\omega}}\)

(1,2,4,5,7)=\({\varepsilon_{\varepsilon_0}}\)

(1,2,4,6)=\({\psi_0({\Omega})}\)

(1,2,4,6,4)=\({\psi_0({\Omega+1})}\)

(1,2,4,6,4,6)=\({\psi_0({\Omega×2})}\)

(1,2,4,6,5)=\({\psi_0({\Omega×{\omega}})}\)

(1,2,4,6,6)=\({\psi_0({\Omega^2})}\)

(1,2,4,6,7)=\({\psi_0({\Omega^{\omega}})}\)

(1,2,4,6,8)=\({\psi_0({\Omega^{\Omega}})}\)

(1,2,4,7)=\({\psi_0({\psi_1(0)})}\)

(1,2,4,7,11...)=\({\psi_0({\Omega_{\omega}})}\)

ここまで-1-HSと同じ

(1,2,4,8)=(1[1[1,2]])=\({\psi_0({\Omega_{\omega}})}\)

(1,2,4,8,4)=\({\psi_0({\Omega_{\omega}}+1)}\)

(1,2,4,8,4,6)=\({\psi_0({\Omega_{\omega}}+{\Omega})}\)

(1,2,4,8,4,7)=\({\psi_0({\Omega_{\omega}+{\psi_1(0)}})}\)

(1,2,4,8,4,8)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}})}})}\)

(1,2,4,8,5)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}})×{\omega}}})}\)

(1,2,4,8,6)=\({\psi_0({\Omega_{\omega}}+{\psi_1({\Omega_{\omega}})×{\Omega}})}\)

(1,2,4,8,6,10)=\({\psi_0({\Omega_{\omega}+psi_1({\Omega_{\omega}})^2})}\)

(1,2,4,8,7)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}+1})}})}\)

(1,2,4,8,7,9)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}+{\Omega}})}})}\)

(1,2,4,8,7,10)=(1,2|4,8,7|9,13,12|14,18,17|...)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}+{\Omega_2}})}})}\)

(1,2,4,8,7,11)=\({\psi_0({\Omega_{\omega}}+{\psi_1({\Omega_{\omega}+{\psi_2(0)}})})}\)

(1,2,4,8,7,12)=\({\psi_0({\Omega_{\omega}+{\psi_1({\Omega_{\omega}+{\psi_2({\Omega_{\omega}})}})}})}\)

(1,2,4,8,8)=(1,2|4,8|7,12|11,17|...)=\({\psi_0({\Omega_{\omega}×2})}\)=B(0,0,0)(1,1,1)(1,1,1)

他

N→Z変換

　項を0の列に、セパレータを1に変換。

計算例

\((3,3,0,2,1)→(0,0,0,0,1,0,0,0,0,1,0,1,0,0,0,1,0,0)\)

前例（推測）

• 巨大数探索スレッド10の>>638
• 巨大数探索スレッド11.75の>>1

S→Z変換

セパレータを0に変換。

計算例

\((0,1,2,3,1,2,2)→(0,1,0,2,0,3,0,4,0,2,0,3,0,3)\)

\((0,0,0)(1,1,1)(2,2,0)(3,1,1)→(0,1,1,1,0,2,2,2,0,3,3,1,0,4,2,2)\)

前例

不明

S→Z/N→O変換

セパレータを0に、項を1の列に変換。

計算例

\((2,2,1,1,1,0,3)→(1,1,1,0,1,1,1,0,1,1,0,1,1,0,1,1,0,1,0,1,1,1,1)\)

セパレータを先頭に置くことでS型にすることができる。

前例

不明

• \(SpS+SZNO=pS/BW\)

N→B変換

\(n→n,n+1\)

計算例

\((0,1,2)→(0,1,1,2,2,3)\)

前例

• \(LPpS=pS+NB\)
• \(m-DS=pS+NB×(m+2)\)

N→L変換

\(n→0,1,2...n\)

計算例

\((3,1,2)→(0,1,2,3,0,1,0,1,2)\)

計算例

• \(SPpS=SpS+NL\)
• \(PpS=pS+NL\)

階差表記

• 初項と階差を使って数列を書き表す。

例:(1,2,3,5,6,8,11)=(1[1,1,2,1,2,4])

​

• 数列システムを作るのに使う場合、0以下の項を含めないほうが便利である(多分)。

例:(1,2,4,5,5,7,10)=(1[1,2,1],5[2,3])

(1[1[1[...]]])の一般項は\(2^n\)

拡張階差表記

• \(a_1[a_2,a_3...]_1=a_1[a_2,a_3...]=a_1,a_1+a_2,a_1+a_2+a_3...\)
• \(a_1[a_2,a_3...]_{m+1}=a_1[(a_1+a_2)[(a_1+a_2+a_3)[...]_m]_m]_m\)

例:\(1[1,1,1]_3=1[2[3[4]_2]_2]_2=1[2[3[7]_1]_2]_2=1[2[3,10]_2]_2=1[2[5[15]_1]_1]_2=1[2[5,20]_1]_2=\)

　 \(1[2,7,27]_2=1[3[10[37]_1]_1]_1=1[3[10,47]_1]_1=1[3,13,60]_1=1,4,17,77\)

\((1[1]_1)=(1,2)\)

\((1[1[1]_2]_1)=(1,2,5)\)

\((1[1[1[1]_3]_2]_1)=(1,2,5,16)\)

\((1[1[1[1[1]_4]_3]_2]_1)=(1,2,5,16,68)\)

\((1[1[1[1[1[1]_5]_4]_3]_2]_1)=(1,2,5,16,68,400)?\)

一般項は不明

極限

\(Max(A)\)を\(A\)の最大の要素とし、\(\text{Row}(A)\)を\(A\)の行数とする。

数極限

システム\(X\)の標準形の列\(A\)と\(B\)について\(Max(A)>Max(B)⇒A>B\)が成り立つとき、

\(X\)は数極限である。

行極限

システム\(X\)の標準形の行列\(A\)と\(B\)について\(\text{Row}(A)>\text{Row}(B)⇒A>B\)が成り立つとき、

\(X\)は行極限である。

例

数極限のシステム：ベクレミシェフの虫、原始数列

数極限でないシステム：大1次数列、バシク行列

行極限のシステム：バシク行列

小極限

システム\(X\)での極限順序数\({\alpha}\)に対応する標準列が\({\Delta}=0\)であるとき、

\(X\)において\({\alpha}\)は小極限である。

例

PS:\((0,0)(1,1)(2,1)(3,0)={\varphi({\omega},0)},Δ=(0,0)⇒\)PSにおいて\({\varphi({\omega},0)}\)は小極限である

大極限

システム\(X\)での極限順序数\({\alpha}\)に対応する標準列が\({\Delta}≠0\)であるとき、

\(X\)において\({\alpha}\)は大極限である。

例

PLPS:\((0,0)(1,1)={\varphi({\omega},0)},Δ=(1,0)⇒\)LPPSにおいて\({\varphi({\omega},0)}\)は大極限である

S型

\((0,1,2,...n)\)を極限とする数列システムをS型とする。

例:小偽原始数列、原始数列、大偽原始数列、大SZNO数列

W型

\((n)\)を極限とする数列システムをW型とする。 例:小1次数列、ベクレミシェフの虫

RN変換

行極限の行列から数極限の行列への変換

\((a_1,a_2,...a_k)→(1,...(a_1個)...1)+(1,...(a_2個)...1)+...(1,...(a_k個)...1)\)

\((0)(1)(2)(3)→\begin{pmatrix}0&1&1&1\\0&0&1&1\\0&0&0&1\end{pmatrix}\)

\(\begin{pmatrix}0&1&2&3\\0&1&1&1\end{pmatrix}→\begin{pmatrix}0&2&2&2\\0&0&1&1\\0&0&0&1\end{pmatrix}\)

\(\begin{pmatrix}0&1&2&3\\0&1&2&3\end{pmatrix}→\begin{pmatrix}0&2&2&2\\0&0&2&2\\0&0&0&2\end{pmatrix}\)

\(RN^{-1}=RN?\)